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Involutions with 1- or 2-dimensional fixed point sets on orientable torus bundles over a 1-sphere and… Holzmann, Wolfgang Herbert 1984

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INVOLUTIONS WITH 1- OR 2~DIMENS IONAL FIXED POINT SETS ON ORIENTABLE TORUS BUNDLES OVER A 1-SPHERE AND ON UNIONS OF ORIENTABLE TWISTED I-BUNDLES OVER A KLEIN BOTTLE by WOLFGANG HERBERT HOLZMANN B .A. , U n i v e r s i t y of C a l g a r y , 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard THE UNIVERSITY OF BRITISH COLUMBIA March 1984 © Wolfgang Herber t Holzmann, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r equ i r ements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Co lumb i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copy i ng o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s unde r s tood t h a t copy ing o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l owed w i t hou t my w r i t t e n p e r m i s s i o n . Department o f Mathemat ics The U n i v e r s i t y o f B r i t i s h Co lumbia 1956 Main Mall Vancouve r , Canada V6T 1Y3 Date A p r i l 2 3 r d r 19B4 T h e s i s S u p e r v i s o r ; D r . E rha rd L u f t A b s t r a c t We o b t a i n a complete e q u i v a r i a n t t o rus theorem fo r i n v o l u t i o n s on 3-manifo lds M. M i s not r e q u i r e d to be o r i e n t a b l e nor i s H,(M) r e s t r i c t e d to be i n f i n i t e . The proof proceeds by a surgery argument. S i m i l a r theorems are g i ven fo r a n n u l i and f o r d i s c s . These are used to c l a s s i f y i n v o l u t i o n s on v a r i o u s spaces such as o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Next we r e s t r i c t our a t t e n t i o n to o r i e n t a b l e t o rus bundles over S 1 or un ions of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . The e q u i v a r i a n t t o rus theorem i s a p p l i e d to the problem of de te rm in ing which of these spaces have i n v o l u t i o n s wi th 1-dimensional f i x e d p o i n t s e t s . I t i s shown tha t the f i x e d p o i n t set must be one, two, t h r e e , or four 1-spheres. Ma t r i x c o n d i t i o n s that determine which of these spaces have i n v o l u t i o n s wi th a g i ven number of V-spheres as the f i x e d p o i n t s e t s are o b t a i n e d . The i n v o l u t i o n s w i th 2-dimens iona l f i x e d p o i n t se t s on o r i e n t a b l e t o rus bundles over S 1 and on un ions of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e are c l a s s i f i e d . Only the o r i e n t a b l e f l a t 3-space forms M, , M 2 and M 6 have i n v o l u t i o n s w i th 2-d imens iona l f i x e d s e t s . Up to con jugacy , M, has two i n v o l u t i o n s , M 2 has four i n v o l u t i o n s , and M 6 has a unique i n v o l u t i o n . i i i Tab le of Contents A b s t r a c t i i L i s t of f i g u r e s v Acknowledgement v i I n t r o d u c t i o n 1 Chapter I. E q u i v a r i a n t T r a n s v e r s a l i t y and D i sc Theorems. . . . 4 <|>1. P r e l i m i n a r i e s 4 §2. E q u i v a r i a n t T r a n s v e r s a l i t y 8 I I . I n v o l u t i o n s on the 3-Ce l l and the S o l i d Torus . . . 25 #3. Some I n v o l u t i o n s 25 #4. I n v o l u t i o n s on the S o l i d Torus 31 I I I . E q u i v a r i a n t Annulus and Torus Theorems 42 #5. Annulus Theorems 42 E q u i v a r i a n t Torus Theorem 48 IV. I n v o l u t i o n s on O r i e n t a b l e I-Bundles Over T o r i and K l e i n B o t t l e s 62 #7. I n v o l u t i o n s on the T r i v i a l I-Bundle Over a Torus 62 #8. I n v o l u t i o n s on the O r i e n t a b l e I-Bundle Over a K l e i n B o t t l e 76 i v V . I n v o l u t i o n s on O r i e n t a b l e Torus Bundles Over a 1-Sphere and on Unions of O r i e n t a b l e Tw is ted I-Bundles Over K l e i n B o t t l e s 90 #9. I n v o l u t i o n s With 1-Dimensional F i x e d Sets . . 90 #10. I n v o l u t i o n s With 2-Dimensional F i x e d Sets . 110 B i b l iography 117 L i s t of F i g u r e s F i g u r e 1 6 F i g u r e 2 • . . 9 F i g u r e 3 24 F i g u r e 4 27 F i g u r e 5 32 F i g u r e 6 43 F i g u r e 7 50 F i g u r e 8 53 F i g u r e 9 55 F i g u r e 10 63 F i g u r e 11 78 F i g u r e 12 99 F i g u r e 13 100 Acknowledgement I wish to thank my s u p e r v i s o r , D r . E rha rd L u f t , whose guidance and encouragement have been i n v a l u a b l e to me.. A l s o I wish to express my g r a t i t u d e to the N a t u r a l S c i ences and E n g i n e e r i n g Research C o u n c i l of Canada and the U n i v e r s i t y of B r i t i s h Columbia fo r p r o v i d i n g me wi th f i n a n c i a l s u p p o r t . I a p p r e c i a t e the e f f o r t s of numerous o the r persons who i n v a r i o u s ways c o n t r i b u t e d to the comp le t i on of t h i s p r o j e c t . With S inan S e r t o z ' s h e l p f u l sugges t i ons the numerous problems encountered in p roduc ing t h i s t h e s i s on a computer were overcome. I wish to thank my pa ren t s f o r the h e l p tha t they have g i ven me in many ways. INTRODUCTION We i n v e s t i g a t e the problem of c l a s s i f y i n g i n v o l u t i o n s wi th 1-dimensional or 2-d imens iona l f i x e d se t s on o r i e n t a b l e t o rus bundles over S 1 or on un ions of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . C u t t i n g these spaces on i n c o m p r e s s i b l e t o r i T g i v e s t r i v i a l I-bundles over t o r i or o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . In Chapter III we prove a complete e q u i v a r i a n t t o rus theorem fo r i n v o l u t i o n s . T h i s theorem a l l ows the c u t t i n g to be done in a manner tha t r e s p e c t s both the i n v o l u t i o n i and the f i x e d set F i x , i . e . , such tha t e i t h e r i T n T = 0 or iT = T and T and F i x are t r a n s v e r s a l . The problem then reduces to one of c l a s s i f y i n g i n v o l u t i o n s on t r i v i a l I-bundles over t o r i or o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e . Such theorems have been proved in [8] and [14] but under a d d i t i o n a l h ypo theses , such a s , the f i r s t homology be ing i n f i n i t e and the m a n i f o l d be ing o r i e n t a b l e . In [11] an e q u i v a r i a n t t o rus theorem was proved under the assumpt ion tha t the f i x e d set i s a number of i s o l a t e d p o i n t s . Our theorem extends these r e s u l t s . In the n o n o r i e n t a b l e case we may have to a l l ow the c u t t i n g t o rus to be r e p l a c e d by a K l e i n b o t t l e . Even in the o r i e n t a b l e case two types of e x c e p t i o n a l cases are p o s s i b l e . To prove the e q u i v a r i a n t t o rus theorem cut and pas te t e chn iques are used . An e q u i v a r i a n t t r a n s v e r s a l i t y theorem i s a l s o r e q u i r e d . In the two d imens iona l case when M i s n o n o r i e n t a b l e " t r a n s v e r s a l i t y can not be gua ran teed . C e r t a i n i n t e r e s t i n g e x c e p t i o n a l p o i n t s a r i s e ; these w i l l be c a l l e d sadd le p o i n t s . Saddle p o i n t s must be t r e a t e d s e p a r a t e l y in the su rgery arguments . Analogous to the t o rus theorem are the annulus and d i s c theorems. These theorems are used in Chapter IV to c l a s s i f y the i n v o l u t i o n s on the t r i v i a l I-bundle over a t o rus and on the o r i e n t a b l e I-bundle over a K l e i n b o t t l e . Let M denote an o r i e n t a b l e t o rus bundle over S 1 or a un ion of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . In [ 6 ] Kim and Sanderson have c l a s s i f i e d the o r i e n t a t i o n r e v e r s i n g i n v o l u t i o n s on o r i e n t a b l e t o rus bund les over S 1 . Our t e chn iques a l low us to c l a s s i f y the i n v o l u t i o n s wi th 2 -d imens iona l f i x e d p o i n t set on a l l M. A s u b c l a s s of these spaces are the o r i e n t a b l e f l a t 3-space forms M, , M 2 , • • ' M 6 , see Wolf [ 1 5 ] . M 6 i s not a t o rus bundle and H,(M 6 ) i s f i n i t e . We show in £ 1 0 tha t M, , M 2 and M 6 are the on ly M hav ing i n v o l u t i o n s w i th 2 -d imens iona l f i x e d p o i n t s e t s . These i n v o l u t i o n s are determined up to con jugacy . The case of 1 -d imens iona l i n v o l u t i o n s on these spaces i s f a r l e s s r e s t r i c t i v e . Each of the o r i e n t a b l e space forms has i n v o l u t i o n s wi th 1 -d imens iona l f i x e d p o i n t se t s but these are not the on ly M w i th such i n v o l u t i o n s . We determine i n *)9 which spaces M have i n v o l u t i o n s wi th 1 -d imens iona l f i x e d p o i n t s e t s . We do not dea l w i th the problem of un iqueness in t h i s t h e s i s . S e v e r a l t o p i c s fo r f u r t h e r r e s e a r c h p resen t t hemse l ves . For example, c l a s s i f y the i n v o l u t i o n s on the n o n o r i e n t a b l e t o rus bund les over S 1 . Can an e q u i v a r i a n t theorem be proved fo r s u r f a c e s of h igher genus? One c o u l d a l s o i n v e s t i g a t e how the r e s u l t s would g e n e r a l i z e from i n v o l u t i o n s to n - c y c l i c a c t i o n s and f i n i t e group a c t i o n s . I. EQUIVARIANT TRANSVERSALITY AND DISC THEOREMS #1. P r e l i m i n a r i e s Use n , u and c to denote set i n t e r s e c t i o n , union and subse t . |_j does not denote d i s j o i n t u n i o n . Use upper i n d i c e s to i n d i c a t e d imens ion . Throughout we use the p i e cew i se l i n e a r c a t e g o r y . T h i s i s to a v o i d w i l d f i x e d se t s which can a r i s e in the t o p o l o g i c a l c a t e g o r y , see M J . A p i e cew i se l i n e a r homeomorphism w i l l be c a l l e d an i somorph ism. Def i n i t ion 1.1 Let M be a m a n i f o l d wi th boundary 9M and F a subman i fo ld of M of lower d imens ion . F i s p roper i f F n 3 M = 3 F . In p a r t i c u l a r a po in t i s proper in M on l y i f i t i s in the i n t e r i o r of M. We w i l l assume tha t a l l subman i fo lds are p r o p e r . F w i l l u s u a l l y denote a s u r f a c e , a compact connected m a n i f o l d . A s u r f a c e F -in a 3-manifo ld M i s i n c o m p r e s s i b l e i f F i s not a 2-sphere or 2 - c e l l and i f f o r each 2 - c e l l B in M wi th B |-| F = 3B there i s a 2 - c e l l D c F w i th 3D=3B. A m a n i f o l d M i s i r r e d u c i b l e i f each 2-sphere in M bounds a 3-ce l l in M. Let M be a connected compact 3-man i fo ld . An i n v o l u t i o n 2 i i s an isomorphism wi th i * i d and i = i d . Let F i x denote the f i x e d set F i x= f i x ( c ) = {x : i (x)=x}. Let i be an i n v o l u t i o n on a m a n i f o l d M and i ' an i n v o l u t i o n on a m a n i f o l d M ' . i and i ' are conjugate i f there i s an isomorphism h:M >M' w i th i ' = h . i . h 1 . C a l l h a c o n j u g a t i o n between i and t ' . L i s c o n j u g a t i o n extendable i f g i ven any con jugate t' of i and an isomorphism h 0 : 9 M >9M' w i th 1 ' I 9 M ' = h ° " 1 I 9 M c h ° 1 t n e n the re i s a c o n j u g a t i o n h:M >M' ex tend ing the isomorphism h 0 . Note tha t i f t i s c o n j u g a t i o n ex tendab le then so i s any con juga te c ' . F u r t h e r , to show c o n j u g a t i o n e x t e n d a b i l i t y i t s u f f i c e s to check the case i ' = t. i i s c o n j u g a t i o n extendable with r e s p e c t to a c l a s s H of isomorphisms 9M >9M', depending on the c h o i c e of con jugate i ' of t, i f at l e a s t f o r any h 0 eH with i ' | = h 0 • i | • ( h 0 1 ) there i s a c o n j u g a t i o n h:M >M' ex tend ing the isomorphism h 0 . The f o l l o w i n g c o n s t r u c t i o n w i l l be used o f t e n . See F i g u r e 1. Let M = B u i B w i th B n iB= 9B n 9 iB and s i m i l a r l y f o r M' and B' . Le t h 0 : B >B' be an isomorphism such tha t h 0 | : B n i B > B ' n t ' B ' i s a c o n j u g a t i o n between i | B n i B and i' |B' n i'B'. h 0 i s extended by e q u i v a r i a n c e to h:M >M' i f we d e f i n e h |_=h 0 and hi _ = i ' . h 0 « ( i M. Then h i s a c o n j u g a t i o n between t|M and t' JM'. 6 t F i g u r e 1. Lemma 1.2 G iven a s i m p l i c i a l s u b d i v i s i o n K of M and an i n v o l u t i o n t of M the re i s a s u b d i v i s i o n L of K wi th t s i m p l i c i a l w i th r e spec t to L. P r o p o s i t ion 1.3 Let t be an i n v o l u t i o n on a m a n i f o l d M. Let L be a s u b d i v i s i o n of M wi th i :L >L s i m p l i c i a l and l e t L' be the f i r s t b a r y c e n t r i c s u b d i v i s i o n of L. Then F i x = f i x ( t ) i s a subcomplex of L ' . F i x i s the union of d i s j o i n t 0-, 1- and 2-d imens iona l proper subman i f o l d s . 0 1 2 Wri te F i x , F i x and F i x r e s p e c t i v e l y f o r the un ions of the 0-, 1- and 2-dimens iona l components of F i x . 0 2 I f v e F i x L J F I X then t i s l o c a l l y o r i e n t a t i o n r e v e r s i n g at v . I f v e F i x 1 then t i s l o c a l l y o r i e n t a t i o n p r e s e r v i n g at v . In p a r t i c u l a r i f M i s o r i e n t a b l e then t i s o r i e n t a t i o n 0 2 r e v e r s i n g i f F i x j j F i x * 0 and t i s o r i e n t a t i o n p r e s e r v i n g i f F i x 1 * 0. 7 P r o o f : Use the f o l l o w i n g : 1) Let A be a s tandard m-simplex (with s tandard s u b d i v i s i o n ) i n v a r i a n t under i . Then F i x n A i s a subcomplex of the f i r s t b a r y c e n t r i c s u b d i v i s i o n of A. 2) If F i x c o n t a i n s a 3-simplex then i = i d . I f v e F i x i s a ve r tex of i n t ( L ) c o n s i d e r the l i n k Lk of v . 3) If L & n F i x c o n t a i n s a 1-ce l l then LknFix i s one 2 1-sphere. So veF ix . 4) If L / c r-|Fix c o n s i s t s of m>0 v e r t i c e s then xUk/i) - m = i (\(Lk) - m) S ince Lk/i i s a su r f a ce and Lk i s a 2-sphere i t f o l l o w s m=2 and hence veF ix 1 . , or i t f o l l o w s m=0 and hence veFix**. QED C o r o l l a r y 1.4 Let t be an i n v o l u t i o n on M wi th f i x e d set F i x . Then M/t = M/(m~i(m) fo r a l l meM) i s a m a n i f o l d w i th p o s s i b l e s i n g u l a r i t i e s . F i x = F i x / i a d i s j o i n t union of subman i fo lds Fix**, F i x 1 and F i x 2 w i th Fix** and F i x 1 proper 2 2 in M/t and wi th F i x a subman i fo ld of 3(M/i) = ( 9 M ) / i u F i x . P r o o f : Cons ide r the l i n k Lk of v e r t i c e s of F i x . QED Remark 1.5 The f i x e d po in t f r e e i n v o l u t i o n s on a m a n i f o l d M co r r e spond to 2- fo ld c o v e r i n g s by M. If i i s an i n v o l u t i o n then M/i=M/(m~c(m) f o r a l l meM) i s 2- fo ld cove red by M. C o n v e r s e l y , i f p:M >X i s a 2- fo ld cover then d e f i n e an i n v o l u t i o n i by r e q u i r i n g t to i n te r change the two p o i n t s of p 1 ( x ) f o r every xeX. t i s the n o n t r i v i a l deck t r a n s f o r m a t i o n induced by p. I n v o l u t i o n s on M w i th f i x e d set F i x co r r e spond to 2- fo ld c o v e r i n g s by M branched on F i x . t| M-Fix i s f i x e d p o i n t f r ee and p| M-Fix i s unbranched. §2. E q u i v a r i a n t T r a n s v e r s a l i t y In o rder to be ab l e to per form s u r g e r i e s on a s u r f a c e F 0 in a 3-manifo ld M we would l i k e to per form an ambient i s o t o p y on F 0 such that the i s o t o p i c s u r f a c e F has the p r o p e r t y tha t F, iF and F i x a re p a i r w i s e t r a n s v e r s a l . T h i s 2 can be done i f the m a n i f o l d i s o r i e n t a b l e . I f F i x *0 and M i s n o n o r i e n t a b l e , however p a i r w i s e t r a n s v e r s a l i t y i s not p o s s i b l e in g e n e r a l . T h i s n e c e s s i t a t e s u s i n g a somewhat weaker form of t r a n s v e r s a l i t y . Lemma 2.1 Let F be a proper s u r f a c e in a 3-mani fo ld w i th F, iF and Fix p a i r w i s e t r a n s v e r s a l . Then the components of F n i F are 1-spheres and proper 1 - c e l l s . I f C i s a component of 2 2 F n c F w i th C n F i x *0 then C L Fix . P r o o f : The f i r s t statement f o l l o w s by t r a n s v e r s a l i t y of F and i F . The second statement f o l l o w s on c o n s i d e r i n g the s t a r 2 of a p o i n t in C n F i x . QED For the 3-ce l l B 3 = { ( x , y , z ) : |x | <1 , | y |<1 , |z |<1} in R 3 l e t i : B 3 >B3 be the map i ( x , y , z ) = ( - x , y , z ) . Then Fix( i ) i s the i n t e r s e c t i o n of B 3 w i th the y z p l a n e . Let S be the 1-sphere ob ta ined as the j o i n of {(1 , 1 ,1) , (-1,-1,1 )} w i th {(-1 ,1 ,-1) , (1f-1,-1)} and l e t D be the cone from (0 ,0 ,0 ) on S. D i s a saddle shaped r e g i o n . See F i g u r e 2. (We c o u l d a l t e r n a t e l y take D d e f i n e d by {z=xy/Vx•x+y-y } u { (0 ,0 ,0 ) } . ) y z 10 N o t i c e tha t D n i D i s the pa r t of the x and y a x i s in B 3 whi le D n F i x ( i ) i s pa r t of the y a x i s . D and F i x ( i ) are t r a n s v e r s a l and iD and F i x ( i ) a re t r a n s v e r s a l , but D and iD are not t r a n s v e r s a l at ( 0 , 0 , 0 ) . There i s a s u b d i v i s i o n making these spaces s i m p l i c i a l w i th a l l the v e r t i c e s on 3B 3 u ( 0 , 0 , 0 ) . D e f i n i t i o n 2.2 Let F be a proper su r f a ce i n a 3-mani fo ld M and i an i n v o l u t i o n on M with f i x e d set F i x . C a l l a p o i n t v a sadd le 2 p o i n t i f v e F n F i x and i f (F, i F , F i x ) n s t a r ( v ) i s i somorphic to ( D , i D , F i x ( i ) ) . Remark 2.3 Saddle p o i n t s e x i s t s i n ce i i s an i n v o l u t i o n wi th f i x e d set F i x ( i ) . A l though 3D, 9 iD , 9 F i x ( i ) are pa i rw i s e t r a n s v e r s a l there i s no 2 - c e l l E w i th 9E=9D and E, i E , F i x ( i ) p a i r w i s e t r a n s v e r s a l . O the rw i se , s i n c e 9 E n 9 i E -F i x ( i ) = ( ± 1 , 0 , 0 ) the re i s a 1-ce l l I of E n i E wi th ( 1 , 0 , 0 ) e 9 l and t h i s 1-ce l l must meet F i x , c o n t r a d i c t i n g the p r e v i o u s lemma. Let d denote the i d e n t i f i c a t i o n ( x , 1 , z ) ~ ( x , - 1 , - z ) for a l l x and z . Then D/d i s an annulus in a s o l i d K l e i n b o t t l e B 3 /d and no i so topy of D/d moves i t to an annulus with F, i F , and F i x ( i ) / d p a i r w i s e t r a n s v e r s a l . 11 D e f i n i t i o n 2.4 Let F be a proper s u r f a c e in a 3-mani fo ld and t an i n v o l u t i o n on M with f i x e d set F i x . Then F, cF, and F i x are a lmost p a i r w i s e t r a n s v e r s a l i f : 1) F, cF and F i x are p a i r w i s e t r a n s v e r s a l except at a f i n i t e number of sadd le p o i n t s , and 2) The on l y components of F n F i x c o n t a i n i n g saddle p o i n t s are 1-spheres and each such 1-sphere c o n t a i n s at most one sadd le p o i n t . Let E be the c l o s u r e of ( F n i F ) - F i x . E c o n s i s t s of d i s j o i n t 1-spheres and proper 1 - c e l l s : in a ne ighborhood of 2 a sadd le p o i n t , F n i F - F i x co r r e sponds to [ - 1 , 0 ) x O x O u ( 0 , 1 ] x O x O in the B 3 model f o r sadd le p o i n t s . Let E be a component of E tha t c o n t a i n s a sadd le p o i n t v . Then E has a f i x e d p o i n t and i s i n v a r i a n t under , i . T h e r e f o r e , e i t h e r E i s a 1-ce l l w i th no f i x e d p o i n t s o ther than v or E i s a 1-sphere wi th e x a c t l y two f i x e d p o i n t s v 1 2 and w. By t r a n s v e r s a l i t y w i s in F i x or F i x . In the l a t t e r case w i s a sadd le p o i n t . We o b t a i n the f o l l o w i n g p r o p o s i t i o n : 12 P r o p o s i t i o n 2.5 Let F, LF and F i x be almost pa i rw i se t r a n s v e r s a l . Then the components of F n i F are of one of the f o l l o w i n g forms: 1) Components wi th no sadd le p o i n t s ( s tandard components ) : 2 a) proper 1-ce l l I w i th I n F i x = 0 or I r- F i x b) proper 1-ce l l I w i th I n F i x = I n F i x * = v , v a p o i n t 2 c) 1-sphere S wi th S n F i x = 0 or S r- F i x d) 1-sphere S wi th S n F i x = S n F i x * = v , u v 2 where v , and v 2 are p o i n t s . 2) Components wi th sadd le p o i n t s : 2 Type I component: S , u I w i th S , n I =Fix n I =w, S, r- F i x and w i s the on l y sadd le p o i n t on S , u I. . 2 Type II component: S 1 U S wi th S i n S = w , S, £ F i x , S n F ix=v u w, v eF ix and w i s the on ly saddle p o i n t on S 1 U S . Type III component: S 1 U S 2 U S w i th S i n S 2 = 0 , S i n S = w^ S i 2 r- F i x , Sr-]Fix= w 1 L J w 2 and w, and w2 are the on ly sadd le p o i n t s on S 1 L J S 2 L j S . Here S, S, and S 2 are 1-spheres, I are 1-ce l l s and w^  are p o i n t s . Note a r e g u l a r ne ighborhood of any of S, S , , or S 2 i s a s o l i d K l e i n b o t t l e in case 2 ) . Thus case 2) does not occur i f the m a n i f o l d i s o r i e n t a b l e . P r o o f : If the r e g u l a r ne ighborhood N of S i s a s o l i d t o rus then F i x n N , F n N and c F n N are a l l a n n u l i or a l l Mobius 13 bands. Cons ide r the components of F n 9N and i F n 3 N in 9N F i x fo r a c o n t r a d i c t i o n . For example, i f they are a l l a n n u l i then l e t A be a component of 9N-F ix . A[-|Fix and A n i F i x are two 1-spheres tha t i n t e r s e c t t r a n s v e r s a l l y at one p o i n t . T h i s i s not p o s s i b l e in an annulus A. Compare wi th the proof of case 3 and 4 in s t ep 1 of the next theorem ( t r a n s v e r s a l i t y theorem) . QED C o r o l l a r y 2.6 If F, tF and F i x are a lmost p a i r w i s e t r a n s v e r s a l , then they are p a i r w i s e t r a n s v e r s a l i f one of the f o l l o w i n g h o l d s : a) M i s o r i e n t a b l e b) F i s a 2 - c e l l c) F i s a annulus w i th 9 F n i 9 F = 0 . P r o o f : In case a) r e g u l a r ne ighborhoods of 1-spheres a re s o l i d t o r i . In cases b) and c) Type II or III components are exc luded s i n ce the 1-sphere S i s n o n s e p a r a t i n g . In case c) type I components are exc luded a p r i o r i , wh i l e in case b) 1-sphere S, sepa ra tes so a proper 1-ce l l C cannot i n t e r s e c t t r a n s v e r s a l l y at one p o i n t . QED 14 A proper 1-cel l I bounds a d i s c D in a s u r f a c e i f I=9D-3F. C o r o l l a r y 2.7 Let F, i F , and F i x be a lmost p a i r w i s e t r a n s v e r s a l and C a proper 1-ce l l or 1-sphere component of F n i F , tha t i s , l e t C be a s t andard component. Then any d i s c in F or iF bounded by C c o n t a i n s only s tandard components. P r o o f : As in case b) in p r e v i o u s c o r o l l a r y . QED E q u i v a r i a n t T ransve rsa 1 i t y Theorem 2.8 Let t be an i n v o l u t i o n on a 3-mani fo ld M. w i th F i x = f i x ( i ) and l e t F 0 be a proper su r f a ce in M. Then the re i s an ambient e- isotopy on M t a k i n g F 0 to a p roper s u r f a c e F such tha t F, tF and F i x are a lmost p a i r w i s e t r a n s v e r s a l . In 3M, i f 3F, i 3 F and F i x are p a i r w i s e t r a n s v e r s a l then the i so topy may be taken to be the i d e n t i t y on 3M-N where N i s a . . , . 2 given ne ighborhood of 3F ix n 3F. P r o o f : Le t F=F 0 be a proper s u r f a c e . By P r o p o s i t i o n 1.3 and Lemma 1.2 subd i v i de M so tha t i i s s i m p l i c i a l w i th r e spec t to the s u b d i v i s i o n , F i x i s a subcomplex of the s u b d i v i s i o n and F i x i s a d i s j o i n t un ion of 0-, 1- and 2-d imens iona l components F i x , F i x and F i x . A l l i s o t o p i e s per formed in the c o n s t r u c t i o n w i l l be done in the s t a r ne ighborhoods of c e r t a i n s i m p l e x e s . By t a k i n g a s u f f i c i e n t l y f i n e s u b d i v i s i o n e - i s o t o p i e s are o b t a i n e d . 2 Step 1) Ad jus t F near F i x . By i s o t o p i e s s i m i l a r to those in the t h i r d s t ep below we can assume F and F i x are t r a n s v e r s a l , the i so topy not moving 9F un l e s s 9F and 9Fix are nont ransver s a l . In p a r t i c u l a r F n F i x ^ 2 = 0. Then F n F i x c o n s i s t s of d i s j o i n t 1-spheres and 1-ce l l components proper in M. 2 Let S be a 1-sphere component of F n F i x . Let N' be a r e g u l a r ne ighborhood of S w i th N't-|F and N ' r- |F ix t r a n s v e r s a l and each an annulus or Mobius band. S has a r e g u l a r ne ighborhood N con t a i ned in i n t ( N ' ) i n v a r i a n t under t w i th no v e r t i c e s on in t (N ) - S such that Nr-]Fix i s a r e g u l a r ne ighborhood of S and F i x n 9 N has a r e g u l a r ne ighborhood Q in 9N which i s i n v a r i a n t under i and has no v e r t i c e s except on F i x |_j 9Q. Case 1) F n N and F i x n N are a n n u l i . Then N i s a s o l i d t o r u s , 3Q has four components and N-Fix c o n s i s t s of two components N, and N 2 which are i n t e r changed by t. Let and J 2 be components of 9Q with J^=N^ and i J ,^ J 2 . Let A^ be the annulus wi th 9A^=J^ U S hav ing no v e r t i c e s except on 9A^. F i s i s o t o p i c to a su r f a ce F' by an ambient i so topy which i s the i d e n t i t y on M-N ' and such tha t F' n N ' n F i x c N a n d F' n N 16 = A , U A 2 . S ince t J ,#J 2 i t f o l l o w s F' n N n t (F ' n N ) = S and F ' n N , i ( F ' n N ) , F i x n N are p a i r w i s e t r a n s v e r s a l . Case 2) F n N and F i x n N are Mobius bands . Then N i s a s o l i d t o r u s , and 9Q has two components tha t are i n t e r changed by t. I f J i s one of t h e s e , then J and S determine a Mobius band A w i th 9A=J. Proceed as in case 1. If M i s o r i e n t a b l e Case 3 and 4 do not a r i s e . Only in these cases do sadd le p o i n t s a r i s e . Case 3) F n N i s an annulus and F i x n N i s a Mobius band. Then N i s a s o l i d K l e i n b o t t l e . Let A be one of the two (open) a n n u l i components of 9N-Fix. There are two 1-spheres J , and J 2 which r ep resen t gene ra to r s of H,(A)=Z w i th J , and J 2 i n t e r s e c t i n g t r a n s v e r s a l l y and at on ly one p o i n t x-. and S bound an annu lus A^ w i th A , n A 2 = S U I where I i s a 1-ce l l w i th 9 I = x u y where yeS . P roceed as in Case 1 u s i n g F ' n N = A , u t A 2 . Then y i s a sadd le p o i n t and F ' n N , t ( F ' n N ) and F i x n N i n t e r s e c t p a i r w i s e t r a n s v e r s a l l y e lsewhere in N. Case 4) F(-|N i s a Mobius band and F i x n N i s an a n n u l u s . T h i s case i s s i m i l a r to case 3. Here A=3N-Q i s an i n v a r i a n t annulus under t. F i n d a curve J tha t bounds a Mobius band by l i f t i n g (from annulus A/ i ) a curve J ' which r e p r e s e n t s twice a genera to r and which i s embedded in A/t except f o r one t r a n s v e r s a l s e l f i n t e r s e c t i o n . 17 When S i s a 1-ce l l component of F n F i x , use an i s o t o p y s i m i l a r to the one of case 1 above. T h i s i s o topy may change 9F in N n 9M. Step 2) Ad jus t F near F i x 1 . By s t ep 1, F p i F i x 1 c o n s i s t s of a number of v e r t i c e s i n i n t ( M ) . I f v e F p j F i x 1 l e t N' be a r egu l a r ne ighborhood of v and l e t N be the s t a r ne ighborhood of v . Take the s u b d i v i s i o n so that N i s in the i n t e r i o r of N ' , F n N i s a proper 2 - c e l l i n N and F i x n N i s a proper 1 - c e l l . S ince F i s t r a n s v e r s a l , F n 9N i s a genera to r of H,(N - F i x ) . Let J ' be a curve in the annulus (9N - F i x ) / i r e p r e s e n t i n g twice a genera tor of t h i s annu lus . Take J ' embedded except f o r one t r a n s v e r s a l s e l f i n t e r s e c t i o n . J ' l i f t s to two 1-spheres J and i J , which on con ing to v g i ve 2 - c e l l s D-and iD. D, iD and F i x are pa i rw i s e t r a n s v e r s a l in i n t N. Proceed as in case 1 of s t ep 1. We o b t a i n a su r f a ce F and a ne ighborhood N of F i x such that F has the r e q u i r e d t r a n s v e r s a l i t y p r o p e r t i e s in N. The f o l l o w i n g c o n s t r u c t i o n a d j u s t s F on l y on s t a r ne ighborhoods of s implexes of F-N where F and iF are not a l r e a d y p a i r w i s e t r a n s v e r s a l . By s u b d i v i d i n g s u f f i c i e n t l y we may assume without l o s s tha t Fix=0. For conven ience assume a l s o 9F=0. Let K be a s u b d i v i s i o n of M wi th c s i m p l i c i a l and F a subcomplex of K. Let A be an m-simplex of F in K wi th m = 0, 1 or 2. De f i ne S t ( A ) , the reduced s t a r of A in K, to be a l l 18 3-simplexes o of K w i th A r- a t oge the r wi th t h e i r f a c e s . Le t S t p ( A ) , the reduced s t a r of A in F, be a l l 2-s implexes a of K wi th A j- o i F toge ther wi th t h e i r f a c e s . Let p:M >M/i be the p r o j e c t i o n . Step 3) There i s a s u b d i v i s i o n of M and a proper s u r f a c e F' e - i s o t o p i c to F such tha t fo r every s implex A of F' e i t h e r p 1 p ( A ) n F ' = A or A i s a 0- or 1-simplex wi th i n t ( S t „ , ( A ) ) and i n t ( S t „ , ( A ) ) t r a n s v e r s a l , r tr C a l l a s implex e x c e p t i o n a l i f i t f a i l s to s a t i s f y these c o n d i t i o n s and i s of the h ighes t p o s s i b l e d imens ion m = 0, 1 or 2. Induct on the number of such s i m p l e x e s . I f there are no e x c e p t i o n a l s implexes the theorem i s e s t a b l i s h e d . Add a l l the v e r t i c e s (and t h e i r t r a n s l a t e s under i ) of form (m+2)/(m+3) b + 1/(m+3) v where b i s the ba r y cen te r of A and v i s a ve r tex of S t (A )-A. T h i s determines a re f inement K' of K w i th the same number of e x c e p t i o n a l s i m p l e x e s ; no m-simplexes are s u b d i v i d e d f o r m=1,2 , wh i l e fo r m=0 t r a n s v e r s a l i t y a l r e ady ho lds away from v e r t i c e s of K. Cons ide r the reduced s t a r s in K ' . 9 S t ' F ( A ) i s a 1-sphere tha t decomposes 9S t ' (A ) i n t o two components D + and D . . There i s an ambient i so topy t a k i n g F to F, = (F - S t ' F ( A ) ) u D + which i s the i d e n t i t y except on S t ' F ( A ) . F, has fewer e x c e p t i o n a l s i m p l e x e s . When m#2 t h i s f o l l o w s s i n ce D + U D . i n t e r s e c t s the i n t e r i o r of any 2-simplex of St (A) t r a n s v e r s a l l y . QED 19 Regular ne ighborhoods of the s tandard components of F|-|tF can be taken in a s p e c i a l fo rm. D e f i n i t i o n 2.9 Let F, iF and F i x be almost pa i rw i s e t r a n s v e r s a l and S a 1-sphere component of F n i F ' Suppose, in a d d i t i o n , tha t the r e g u l a r ne ighborhood of S in F and iF i s an annu lu s . Then the re e x i s t s a r e g u l a r ne ighborhood V c int (M) of S wi th the f o l l o w i n g p r o p e r t i e s : 1) V n F and V n iF a re a n n u l i . S ince these i n t e r s e c t t r a n s v e r s a l l y , V i s a s o l i d t o r u s . 2) F i x and 3V i n t e r s e c t t r a n s v e r s a l l y , F i x n F n V £ S and the c l o s u r e of each component of ( F i x n V ) - S meets S and 3V. In p a r t i c u l a r F i x ° n V = 0 . 3) F i x n V i s an annu lu s , two proper 1-ce l l s or empty. 4) I f tS=S then iV=V 5) I f iS#S then iV n V = 0 and the above p r o p e r t i e s ho ld s i m u l t a n e o u s l y f o r iV . P rope r t y 3) can be ar ranged s i n ce i f F i x n S * 0 then tS=S. i i s an i n v o l u t i o n on a 1-sphere so e i t h e r i= id or c has e x a c t l y two f i x e d p o i n t s . C a l l V a s tandard ne ighborhood of S. The four 1-spheres ( F u i F ) n 3 V decompose 3V i n t o four ( c losed ) a n n u l i a 1 f a 2 , /3, and (S2 w i th a i n a 2 = 0 and /3 i n/3 2=0. C a l l these a n n u l i the 20 s tandard a n n u l i c o r r e s p o n d i n g to the s tandard ne ighborhood of V. Suppose tS=S. R e l a b e l l i n g , i f n e c e s s a r y , we may assume L (a , n j3, ) = ( a 1 n 0 2) . I t f o l l o w s tha t t a ,=a , . Then t /3, = /3 2 and ca 2 = a 2 . When F i x n W 0 we o b t a i n F i x n a 1 ? i 0 , F i x n a 2 ? t 0 , F i x n 0 , =0=Fix n / 3 2 , and each component of F i x meets both a , and a 2 • D e f i n i t i o n 2.10 Let S be a 1-ce l l component of F i x n t F ix where F, i F , F i x are p a i r w i s e t r a n s v e r s a l (near S ) . Then the re e x i s t s a r e g u l a r ne ighborhood V of S w i th V n 9 M a. r egu l a r ne ighborhood of 9S, c a l l e d a s tandard ne ighborhood of S wi th the f o l l o w i n g p r o p e r t i e s : . 1) V n F and V n iF are 2 - c e l l s w i th 9 M n V n F and 9 M n V n i F each two 1 - c e l l s . N e c e s s a r i l y V i s a 3 - c e l l . 2) , 4) and 5) as fo r 1-sphere s tandard ne ighborhoods . 3) F i x n V i s a d i s c , one proper 1-ce l l or empty. The four 1-ce l l s ( F u i F ) n 9V-9M s u b d i v i d e 9V-9M i n t o four d i s c s a , , a 2 , /3, and j32 w i th a , na 2 =0 , /3,n/32=0 and the p r o p e r t i e s as in the p r e v i o u s s i t u a t i o n . C a l l these d i s c s the s tandard d i s c s c o r r e s p o n d i n g to V. 21 Remark 2.11 In the f o l l o w i n g theorem c e r t a i n 1-sphere components S of F n i F have s tandard ne ighborhoods because S bounds d i s c s in F and i F . In the d i s c theorem and p a r t i a l annulus theorem, F i s o r i e n t a b l e so aga in the re are s tandard ne ighborhoods . In the t o rus theorem the c o n s t r u c t i o n w i l l be made so as to keep S in t h i s form a lways . In the annulus theorem the case of a n o n o r i e n t a b l e F, a Mobius band, w i th 1-sphere components i s t r e a t e d s e p a r a t e l y . Theorem 2.12 Let M be a 3-mani fo ld wi th i n v o l u t i o n i and F 0 be an i n c o m p r e s s i b l e proper s u r f a c e . Then the re i s an ambient i s o t o p y of M which i s an e- isotopy on 3M t a k i n g F 0 to a proper su r f a ce F such that F, i F , and F i x are a lmost p a i r w i s e t r a n s v e r s a l and no 1-spheres i n F|-| i F bound 2 - c e l l s in F. I f on 3M, 3F, i3F and 3F ix are p a i r w i s e t r a n s v e r s a l then the i so topy may be taken to be the i d e n t i t y on 3M-N 2 where N i s a g i ven ne ighborhood of 3F ix n ^ F . P r o o f : By the p r e ced ing t r a n s v e r s a l i t y theorem there i s an F w i th a l l the above p r o p e r t i e s except p o s s i b l y 1-spheres in F n i F bound 2 - c e l l s in F. By C o r o l l a r y 2.7 those 2 - c e l l s c o n t a i n no sadd le components. Let S be a 1-sphere of F n i F innermost in i F , tha t i s , there i s a 2 - c e l l D f- /.Fix w i th 22 D n F = 3 D = S . S ince F i s c o m p r e s s i b l e , S bounds a 2 - c e l l B in F. I f iS=S then we may assume iB=D. Let V be a s t anda rd ne ighborhood of S. Such a ne ighborhood e x i s t s s i n c e S bounds a d i s c in F and i F . Let a be the s tandard annulus meet ing D but not B. Then Lar~]Ci=0. There i s a b i c o l l a r D x [ - 1 , l ] of D=DxO with 3DX [ -1 ,1 ] = DX [ - 1 , 1 ] n F = Sx [-1 ,1 ] and wi th Dx1 n a * 0. S ince D i s innermost i t f o l l o w s tha t f o r a s u f f i c i e n t l y t h i n c o l l a r (Dx1 ) n i (Dx1 ) = 0 and F n i(Dxl)=0. Cons ide r F ' = ( F - ( B u S x [ - 1 , 1 ] ) ) u D x 1 . Then F ' n t F ' C ( F n i F ) - S and F ' , i F ' and F i x are a lmost p a i r w i s e t r a n s v e r s a l . S ince M i s i r r e d u c i b l e and D u B i s a 2-sphere , F' and F are ambient i s o t o p i c by an i s o t o p y be ing the i d e n t i t y on 3M. By i n d u c t i o n , a l l 1-spheres bounding 2- c e l l s can be removed. QED Def i n i t i o n 2.13 A 2 - c e l l B in a 3-mani fo ld i s e s s e n t i a l i f i t i s p roper and 3B does not bound a 2 - c e l l i n 3M. In an i r r e d u c i b l e 3- man i f o l d a nonsepa ra t i ng proper 2 - c e l l i s e s s e n t i a l . The f o l l o w i n g theorem a l s o appears in [ 3 ] , 23 D i s c Theorem 2.14 Let M be an i r r e d u c i b l e 3-mani fo ld w i th i n v o l u t i o n i . Suppose M has an e s s e n t i a l 2 - c e l l B 0 . Then the re i s an e s s e n t i a l 2 - c e l l B r- M such tha t B and F i x are t r a n s v e r s a l and e i t h e r B n tB= 0 or tB=B. In the former case B n F i x = 0 and in the l a t t e r case B n F i x i s a proper 1-ce l l of B or one p o i n t in the i n t e r i o r of B. If 3 B o n t 3 B o = 0 then one can take 3B=3B 0 and B and B 0 are ambient i s o t o p i c by an i so topy tha t i s the i d e n t i t y on 3M. P r o o f : By Theorem 2.12 and C o r o l l a r y 2.6 the re i s an e s s e n t i a l 2 - c e l l B w i th B, iB and F i x p a i r w i s e t r a n s v e r s a l , B and B 0 ambient i s o t o p i c and B n i B i s e i t h e r empty or c o n s i s t s of proper 1-ce l l s o n l y . Assume B n i B * 0 ( i n p a r t i c u l a r then 3 B 0 n iBo*0). By i n d u c t i o n i t s u f f i c e s to show how to ob ta in a new 2 - c e l l B^  w i th fewer 1-ce l l s in Let D be an outermost d i s c of B: D c B w i th D n i B=3D r i tB = I a proper 1-cel l of B and 3D-I c 3B. If iI=1 d e f i n e D'=tB-tD. If t l# l d e f i n e D' to be the c l o s u r e of the component of 1B-1D tha t does not c o n t a i n t l . See F i g u r e 3. Let V be a s tandard ne ighborhood of I and l e t a , , a 2 and 0 be s tandard d i s c s of V w i th a,|-|a2=0, a i n/3 n D ^ 0 and j3 n D ' * 0 . Cons ide r B ^ D M / S M D ' ) - i n t (V ) and B 2=D u ( iB-D' ) . F i g u r e 3. Then B i n i B , c ( B n i B ) - i . i f B , i s e s s e n t i a l we are done by i n d u c t i o n or a r r i v e at case B1j-|iB 1=0. I f B, i s not e s s e n t i a l then 9B, bounds a 2 - c e l l E of 9M. S ince M i s i r r e d u c i b l e the 2-sphere B 1 U E bounds a 3 - c e l l . T h i s 3 - c e l l does not meet I, o the rw ise iB would not be e s s e n t i a l . Us ing the 3 - c e l l c o n s t r u c t an ambient i s o t o p y t a k i n g B 2 to i B . So we may assume B 2 i s e s s e n t i a l . I f I= 11 we have i B 2 =B 2 and note tha t F i x n B 2 c F i x n I which i s n e c e s s a r i l y a p o i n t of I or a l l of I. I f l n i l = 0 c o n s i d e r a s u f f i c i e n t l y t h i n b i c o l l a r Dx[-1,-1] of D=DxO such tha t D x [ - 1 , - l ] n I i s a b i c o l l a r of I i n tB and Dx1 meets a , . Then B 2 ' = (Dx1 u i B ) - ( I x [ - 1 , 0 ) U D ' ) i s e s s e n t i a l s i n c e i t i s i s o t o p i c to B 2 and B 2 ' n t B 2 ' c B n iB - I. QED 25 I I . INVOLUTIONS ON THE 3~CELL AND THE SOLID TORUS #3. Some I n v o l u t i o n s -The c l a s s i f i c a t i o n of i n v o l u t i o n s on a s o l i d t o rus w i l l be u s e f u l in the proof of theorems in the next c h a p t e r . The d i s c theorem w i l l be used to reduce the problem of c l a s s i f y i n g i n v o l u t i o n s on a s o l i d t o rus to one of c l a s s i f y i n g the i n v o l u t i o n s on a 3 - c e l l . D e f i n i t i o n 3.1 Let C be the complex numbers. Let I=I 1 =[-1,1] be the s tandard 1-ce l l S 1 = { z e C : |z |= l } be the s t anda rd 1-sphere D 2={zeC : |Z |<1} be the s tandard 2 - c e l l T 2 = S 1 x S 1 be the s tandard t o rus D + ={zeD 2 : z=x+y• i , y^O}, a 2 - c e l l Re={zeD 2 :z=z} j- D 2 , a proper 1-ce l l Im={zeD 2 :z = -z} £ D 2 , a p roper 1-ce l l De f i ne i n v o l u t i o n s on the above spaces as f o l l o w s . On S 1 : K ( Z ) = Z which i s o r i e n t a t i o n r e v e r s i n g w i th f i x e d set two p o i n t s ± 1 . a(z)=-z which i s o r i e n t a t i o n p r e s e r v i n g and f i x e d po in t f r e e . Then a.K=fc .a=-K i s con juga te to K by a r o t a t i o n by 9 0 ° . On D 2 : Jc(z)=z o r i e n t a t i o n r e v e r s i n g wi th f i x e d set one 1-ce l l Re. a(z)=-z which i s o r i e n t a t i o n p r e s e r v i n g w i th f i x e d set one po in t 0. Then -k i s con juga te to ic and has 26 f i x e d set Im. On I : r ( t )=-t which i s o r i e n t a t i o n r e v e r s i n g w i th f i x e d set one p o i n t 0. De f i ne the map p : S 1 x S 1 >S 1 xS 1 by p(z,w)=(zw,w) and map p : D 2 x S 1 >D 2 xS 1 s i m i l a r l y . De f i ne i n v o l u t i o n o r S ^ S 1 >S 1 xS 1 by co (z , w ) = (w , z ) , which has f i x e d set one 1-sphere { ( z , z ) : z } . Lemma 3.2 There are f i v e i n v o l u t i o n s up to con jugacy on an annulus S 1 x l . They a r e : 1) a x i d which i s o r i e n t a t i o n p r e s e r v i n g and f i x e d po in t f r e e , 2) axr which i s o r i e n t a t i o n r e v e r s i n g and f i x e d p o i n t f r e e , 3) K X T which i s o r i e n t a t i o n p r e s e r v i n g wi th f i x e d set two p o i n t s , 4) i d x r which i s o r i e n t a t i o n r e v e r s i n g w i th f i x e d set a 1-sphere, and 5) K x i d which i s o r i e n t a t i o n r e v e r s i n g w i th f i x e d set two proper 1 - c e l l s . P r o o f : When the d imens ion of the f i x e d set i s one the f i x e d set s e p a r a t e s . In the o ther case use the E u l e r c h a r a c t e r i s t i c argument g i ven i n pa r t 4) of proof of P r o p o s i t i o n 1.3. QED 27 D e f i n i t i o n 3.3 For the 3-ce l l D 2 x l d e f i n e the f o l l o w i n g i n v o l u t i o n s (see F i g u r e 4 ) : j 2 = i d x r hav ing f i x e d set a proper 2 - c e l l D 2 x 0 . J , = K X T hav ing f i x e d set an unkot ted 1-ce l l RexO. jo=axr hav ing f i x e d set one p o i n t 0x0. j 2 and jo are o r i e n t a t i o n r e v e r s i n g wh i le i s o r i e n t a t i o n p r e s e r v i n g , j , i's con juga te to j , ' = a x i d which has f i x e d set 0x1 . Theorem 3.4 An i n v o l u t i o n on a 3 - c e l l i s con juga te to j 2 , j i or j 0 . A l l i n v o l u t i o n s on a 3 - c e l l are c o n j u g a t i o n e x t e n d a b l e . P r o o f : Let t be an i n v o l u t i o n on 3-ce l l E. Apply Lemma 1.2 and P r o p o s i t i o n 1.3. 2 2 2 Suppose F i x #0. S ince F i x i s p roper and ^ , £ = 1 , F i x s e p a r a t e s . Let E, and E 2 be the components, E = E 1 U E 2 w i th J2 j i j o F i g u r e 4. 28 F i x = E i n E 2 . I f F i x were compress i b l e then l e t B be a . 2 . compress ing d i s c in E 1 f say . Then iB compresses F i x in E 2 . U s ing a M a y e r - V i e t o r i s sequence we see [3B] must be t r i v i a l 2 2 i n H , ( E , ) © H i ( E 2 ) = H , ( F i x ). Hence F i x i s a proper 2 - c e l l . We show i i s con juga te to j 2 . Let D 1 = D 2 X [ 0 , 1 ] . Cons t ru c t an isomorphism h 0 from the d i s c which i s the c l o s u r e of B E ^ F i x to the c l o s u r e of 3D, - D 2 x 0 . In the c o n j u g a t i o n ex tendab le case we may assume t h i s isomorphism i s g i v e n . Extend h 0 over the f i x e d set F i x and then cone to a p o i n t to o b t a i n an isomorphism h :E , > D 2 X [ 0 , 1 ] , Extend t h i s isomorphism by e q u i v a r i a n c e to get a c o n j u g a t i o n . Suppose F i x =0. The E u l e r c h a r a c t e r i s t i c argument r e f e r r e d to in pa r t 4) of p roo f of P r o p o s i t i o n 1.3 a p p l i e d to i | 3E shows F i x n 3 E has 0 or 2 f i x e d p o i n t s . In the former 1 2 case i i s o r i e n t a t i o n r e v e r s i n g so F i x u F i x =0 and in the l a t t e r case F i x 1 * 0. 1 2 Suppose F i x |_jFix =0. By a L e f s h e t z number argument i has one f i x e d p o i n t o n l y , c a l l i t v . 3E/t i s a p r o j e c t i v e p l ane so there i s a c o n j u g a t i o n h 0 : 3 E > 3(D 2 x I ) between c| and j 0 | . In the c o n j u g a t i o n ex tendab le case h 0 i s g i v e n . By s u b d i v i d i n g we may assume s t a r (v) r-| 3E=0 and E - i n t ( s t a r ( v ) ) = S 2 x l . Extend h to a c o n j u g a t i o n E s t a r ( v ) > D 2 x I - s t a r ( 0 , 0 ) . T h i s can be done by Theorem 1 in [9] s i n c e t has no f i x e d p o i n t on 3E. F i n a l l y cone to v . 29 Suppose F i x #0. By the above F i x n 3 E i s two p o i n t s . Cons ide r the double E U E ' of E. I t i s a 3-sphere w i th i n v o l u t i o n i u i ' induced by t. By a r e s u l t of Waldhausen [16 ] , t h i s i n v o l u t i o n has f i x e d set one unknot ted 1-sphere. Let B be a 2 - c e l l w i th 3B the f i x e d set and such that B i s in gene ra l p o s i t i o n w i th respec t to 3E. B n E i s a punctured 3 - c e l l and a l l but one component of B n 3 E i s a 1-sphere in the i n t e r i o r of B. By s tandard arguments these can be removed g i v i n g a 1-ce l l B' c E w i th 3B' c 3 E u F i x * . T h i s shows F i x 1 i s one unknot ted proper 1 - c e l l . We c l a i m there i s a (nonproper) d i s c B embedded in E w i th F i x £ 3B r- F i x u 3 E such that B n i B = F i x . Moreover , i f C E 3E i s a 1-ce l l w i th C n tC=3C=3Fix then we may assume 3 B n 3 E = C . To e s t a b l i s h t h i s c l a i m note tha t i f N i s a s t a r ne ighborhood of F i x then the c l o s u r e V of E-N i s a s o l i d t o rus wi th i| f i x e d p o i n t f r e e and o r i e n t a t i o n p r e s e r v i n g . V n N i s an annulus and t | ( V n N ) i s a l s o f i x e d p o i n t f r e e . Us ing these f a c t s and D i s c Theorem 2.14 i t i s p o s s i b l e to c o n s t r u c t a d i s c as r e q u i r e d in the c l a i m . Cons t ruc t a c o n j u g a t i o n to j , as f o l l o w s . Cons t ru c t an isomorphism B >D+xO and extend by e q u i v a r i a n c e to B u c B >DxO. In the c o n j u g a t i o n ex tendab le case we set C = h 1 ( ( 3 D n D + )x0) . Then B u t B sepa ra tes E i n t o two 3 - c e l l components. We extend to an isomorphism over one of these components and then by e q u i v a r i a n c e to the o t h e r , g i v i n g a 30 c o n j u g a t i o n E >DxI. QED Lemma 3.5 Let F be a 2 s i ded s u r f a c e in a 3-mani fo ld M and l e t t be an i n v o l u t i o n on M wi th iF=F and such tha t i i n t e r changes s i d e s of F. Then F i s ambient i s o t o p i c to a s u r f a c e F' w i th F' n i F ' = 0 . P r o o f : Cons t ru c t an t i n v a r i a n t b i c o l l a r F X [ - 1 , 1 ] of F=FxO by us ing a s t a r ne ighborhood of F. Then c o n s i d e r F '=Fx1. QED Remark 3.6 Suppose tF=F fo r a 2-s ided s u r f a c e F. Le t N be the 3-mani fo ld ob ta ined by c u t t i n g M a long F. That i s , r ep l a ce F X [ - 1 , 1 ] £ M by d i s t i n c t c o p i e s F ,x [-1 ,0 ]=Fx [-1 ,0 ] and F 2 x [ 0 , 1 ] = F x [ 0 , 1 ] . N has a s u b d i v i s i o n induced from M. Let d : F , = F , x 0 >F 2 =F 2 xO be the c a n o n i c a l i d e n t i f i c a t i o n . Then M = N/d. S ince t i s s i m p l i c i a l the re i s a c a n o n i c a l i n v o l u t i o n k on N w i th t=k/d. Note k.d=d.k and k ( F , ) = F 1 i f f i does not in te rchange the b i c o l l a r . Conve r se l y i f k.d=d.k f o r an i n v o l u t i o n k then k induces an i n v o l u t i o n i in M wi th iF=F. 31 ^4. I n v o l u t i o n s on the S o l i d Torus D e f i n i t i o n 4.1 Let V be the s o l i d t o rus V=D 2 xS 1 = { (z ,w) : | z |<1 , | w | = 1, z,weC}. R e c a l l D e f i n i t i o n 3 .1 . D e f i n e the f o l l o w i n g i n v o l u t i o n s on V (see F i g u r e 5 ) : J A =/cxid hav ing f i x e d set the annulus R e x S 1 . J M = p . ( K x i d ) hav ing f i x e d set the Mobius band { ( s - e 7 r i t , e 2 7 r i t ) : 0 < s < 1 , - 1 < t < l ] J2D=idx/c hav ing f i x e d set two 2 - c e l l s D 2x±1 . J D p=jo. ( i d x K ) ' hav ing f i x e d set a 2 - c e l l and a p o i n t D 2x1 u Ox-1. j s = a x i d hav ing f i x e d set one 1-sphere OxS 1 . J 2 £ = K X K hav ing f i x e d set two 1-ce l l s R e x ± 1 . J2p=ax/c hav ing f i x e d set two p o i n t s 0x± 1 . J N=/cxa f i x e d p o i n t f r ee and o r i e n t a t i o n r e v e r s i n g . J Q = i d x a f i x e d po in t f r e e and o r i e n t a t i o n p r e s e r v i n g . So J M ( z ,w)=(zw,w) and J D P ( z , w ) = ( z w , w ) . The s u b s c r i p t d e s c r i b e s the f i x e d p o i n t set or f o r the f i x e d p o i n t f r ee i n v o l u t i o n s the o r i e n t a b i l i t y t y p e . R e c a l l by P r o p o s i t i o n 1.3, s i n c e V i s o r i e n t a b l e , i n v o l u t i o n s w i th 0- or 2-d imens iona l f i x e d se t s are o r i e n t a t i o n r e v e r s i n g . Those w i th 1-dimensional f i x e d se t s are o r i e n t a t i o n p r e s e r v i n g . None of the above i n v o l u t i o n s are con jugate s i n c e a l l have d i f f e r e n t f i x e d se t s or o r i e n t a t i o n t ype . Us ing V=D 2 xI/d 32 33 where d=ax ( r | 3 I ) , i n v o l u t i o n s con juga te to j and J D p can be d e f i n e d as f o l l o w s : J M '=/cxid/d hav ing f i x e d set Mobius band Rex l /d J D p ' = i d x T / d hav ing f i x e d set D 2 x 0 u 0 x l / d . Theorem 4.2 If i and i ' are i n v o l u t i o n s on V=D 2 xS 1 w i th nonempty i somorph ic f i x e d p o i n t s e t s or i f i and i ' a re f i x e d po in t f r ee and of the same o r i e n t a t i o n t ype , then t and i ' are c o n j u g a t e . An i n v o l u t i o n on V i s con jugate to one of the n ine i n v o l u t i o n s l i s t e d above. P r o o f : Let i be an i n v o l u t i o n on V. We show i t i s con juga te to a s t andard i n v o l u t i o n . For any g iven e s s e n t i a l 2 - c e l l B 0 of V the re i s an isomorphism h of V which takes B 0 to B=D 2 x-1. So by a p p l y i n g the D i s c Theorem 2 .14 , and r e p l a c i n g i by the con juga te i n v o l u t i o n h 1 . i . h f o r a s u i t a b l e h, we may assume i s a t i s f i e s : Case 1) iB=B and B i n t e r s e c t s F i x t r a n s v e r s a l l y at Rex-1 Case 2) iB=B and B i n t e r s e c t s F i x t r a n s v e r s a l l y at Ox-1 or 3') iB nB=0 There i s an isomorphism ( D 2 x l ) / d >D 2 xS 1 where d = i d x ( r | g j ) 1 7T t i s g i ven by d ( z , t ) = ( z ; e ). The isomorphism takes D 2 x - l / d to B. Wr i te a l s o B=D 2 x-1. In case 3') by a d j u s t i n g the isomorphism h we may assume that iB=D 2 x1=(D 2 xO)/d. C a l l C.=D 2 x [-1 ,0 ] and C + = D 2 x [ 0 , l ] . The case 3') s p l i t s i n t o two c a s e s : Case 3) iB n B=0 and iC + =C. Case 4) tB n B=0 and t C + = C + . We show f i r s t , i f the i n v o l u t i o n t f a l l s i n t o : case 1) i t has f i x e d set tha t of J a or J M , case 2) i t has f i x e d set that of j g , case 3) i t i s f i x e d p o i n t f r e e as J N and j Q a r e , case 4) i t has f i x e d set that of J 2 r j ' ^DP' ^2C o r ^2P' and we show second, i f i and t' f a l l i n t o the same case 1) -4) then t and i* are c o n j u g a t e . T h i s w i l l complete the proof because the nine s tandard i n v o l u t i o n s cover a l l p o s s i b l e f i x e d se t s tha t can a r i s e and none occurs in more than one case 1) - 4 ) . A l l c o n s t r u c t i o n s done f o r i are to be performed f o r i ' a l s o even i f not e x p l i c i t l y s t a t e d . Use a prime ' to denote the c o r r e s p o n d i n g c o n s t r u c t . In Case 1): The i n v o l u t i o n t on D 2 x l / d induces an i n v o l u t i o n X on D 2 x l w i th the p rope r t y X.d=d.X when r e s t r i c t e d to D 2 x 3 I . F i x (X ) i s proper and 2-dimens iona l s i n ce F i x = f i x ( i ) i s t r a n s v e r s a l to B. So X i s con juga te to the s t anda rd i n v o l u t i o n j 2 of the 3 - c e l l . In p a r t i c u l a r F i x ( X ) i s a 2 - c e l l . . F i x i s ob t a i ned by i d e n t i f y i n g two d i s j o i n t 1-ce l l s in the boundary of the two c e l l so F i x i s 35 a n a n n u l u s o r M o b i u s b a n d . S u p p o s e i ' i s g i v e n . X | B a n d X | B ' a r e c o n j u g a t e s o t h e r e i s a c o n j u g a t i o n h : B > B ' . U s i n g d a n d d ' w e may e x t e n d h t o a c o n j u g a t i o n D 2 x 9 I > D 2 l x 3 I . 3 F i x ( X ) d e c o m p o s e s 3 D 2 x I i n t o t w o 2 - c e l l s w h i c h a r e i n t e r c h a n g e d u n d e r X . L e t J b e a n ( o p e n ) c o m p o n e n t o f (3D 2 x-1) - F i x ( X ) s e l e c t e d s o t h a t h ( J ) = J ' . T h e n F i x i s a n a n n u l u s i f J a n d d ( J ) a r e i n t h e s a m e 2 - c e l l d e t e r m i n e d b y 3 F i x ( X ) a n d F i x i s a M o b i u s b a n d i f J a n d i d ( J ) a r e i n t h e s a m e 2 - c e l l d e t e r m i n e d b y 3 F i x ( X ) . T h e r e f o r e , h c a n b e e x t e n d e d t o a c o n j u g a t i o n h : D 2 x 3 I u F i x ( X ) > D 2 ' x 3 I u F i x ( X ' ) . E x t e n d h o v e r o n e o f t h e 2 - c e l l s t h a t 3 F i x ( X ) d e c o m p o s e s D 2 x l i n t o . T h e n e x t e n d t o t h e o t h e r c e l l b y e q u i v a r i a n c e . T h i s g i v e s a c o n j u g a t i o n h d e f i n e d o n 3 ( D 2 x I ) . B y t h e c o n j u g a t i o n e x t e n d a b l e p r o p e r t y f o r t h e 3 - c e l l , h e x t e n d s t o a l l o f D 2 x l a n d h e n c e i n d u c e s a c o n j u g a t i o n o n D 2 x l / d b e t w e e n t a n d t ' . I n c a s e 2): A s i n c a s e 1) t h e i n v o l u t i o n i o n D 2 x l / d i n d u c e s a n i n v o l u t i o n X o n D 2 x l w i t h X . d = d . X o n D 2 x 3 I . F i x ( X ) i s p r o p e r a n d 1 - d i m e n s i o n a l s o X i s c o n j u g a t e t o t h e s t a n d a r d i n v o l u t i o n j , o f t h e 3 - c e l l . S o F i x i s a 1 - s p h e r e . S u p p o s e t ' i s g i v e n . X | B a n d X | B ' a r e c o n j u g a t e s o t h e r e i s a c o n j u g a t i o n h : D 2 x 3 l > D 2 , x 3 I w i t h h . d = d ' . h . S i n c e X|3B i s o r i e n t a t i o n p r e s e r v i n g , h e x t e n d s t o 3 ( D 2 x I ) . 36 By the c o n j u g a t i o n ex tendab le p rope r t y f o r 3 - c e l l s , h extends to a l l of D 2 x l and hence induces a c o n j u g a t i o n on D 2 x l / d . In case 3 ) : The i n v o l u t i o n must be f i x e d p o i n t f r e e . Let i and i ' be of same o r i e n t a t i o n t y p e . Cons t ru c t an isomorphism h:B >B' and extend to a c o n j u g a t i o n h : B u i B > B ' u c ' B ' by e q u i v a r i a n c e . S ince the o r i e n t a t i o n type i s the same, h extends to a l l of 9C + and then to an isomorphism h : C + > C ' + by c o n i n g . F i n a l l y extend to D 2 xS 1 =C+|_j C. by e q u i v a r i a n c e . In case 4 ) : t | C + and t | C . a re i n v o l u t i o n s on 3 - c e l l s so each has f i x e d set a p o i n t , a proper 1-ce l l or a proper 2 - c e l l . S ince ( B u i B ) n F i x = 0 i t f o l l o w s F i x=F ix ( i |C + ) u F i x ( t | C . ) . Moreover i | C + and t | C . must have the same o r i e n t a t i o n type so F i x i s one o f : two 2 - c e l l s , two p o i n t s , a 2 - c e l l un ion a p o i n t , or two 1 - c e l l s . Suppose t and c' have i somorph ic f i x e d s e t s . Arrange n o t a t i o n so tha t t | C + and t ' | C + " have i somorph ic f i x e d s e t s . Cons t ru c t a c o n j u g a t i o n h : B u i B > B ' u t ' B ' as in case 3 ) . In view of the c o n j u g a t i o n extendab le p rope r t y of 3 - c e l l s , i t s u f f i c e s to show h extends to a c o n j u g a t i o n 9C + > 3 C + ' . Le t G=3C+ and l e t F i x now denote F i x ( t | C + ) . Case 4.1) i | C + i s con juga te to j 2 . F i x decomposes G i n t o two components one of wh ich , E, c o n t a i n s B. Extend h to an isomorphism h:E >E' and by e q u i v a r i a n c e to a 37 c o n j u g a t i o n h:G >G ' . Case 4.2) i | C + i s con juga te to j , . G/i i s a 2-sphere such that F i x = F i x / i misses B/ i . So l i f t i n g an a p p r o p r i a t e 1-ce l l J , i n (G/t)-(B/t) g i v e s a 1-ce l l J of G wi th J n i J = F i x . In a d d i t i o n , J u t J determines a component E of 3G c o n t a i n i n g B but not iB . De f i ne an isomorphism h 0 : J >J' and extend by e q u i v a r i a n c e to J u t J . N o t i c e we c o u l d have s e l e c t e d h 0 : J > i ' J ' i n s t e a d , so i f o r i e n t a t i o n s are f i x e d fo r J u tJ and J ' | j t ' J ' , we may s e l e c t h 0 to be e i t h e r o r i e n t a t i o n p r e s e r v i n g or o r i e n t a t i o n r e v e r s i n g . Hence h | B U h 0 extends over the (open) annulus E-(B u J u i j ) to an isomorphism h ^ E > E ' . Extend by e q u i v a r i a n c e to G. Case 4.3) i | C + i s con juga te to j 0 . Then E = ( G - i n t ( B u i B ) ) / i i s a Mobius band and h induces an isomorphism 3E >3E ' . T h i s isomorphism extends to a l l of E. G double cove r s G/t and the isomorphism l i f t s to an isomorphism h ' :G >G' ex tend ing h. By c o n s t r u c t i o n h' i s a c o n j u g a t i o n . QED Let C^P be the space ob ta ined by con ing a r e a l p r o j e c t i v e space to a p o i n t v^. Then (C^P,v^) i s i somorph ic to ( B 3 , 0 ) / j o . The d e s c r i p t i o n s of the s tandard i n v o l u t i o n s j on a s o l i d t o rus V can be used to compute V / j . Use F i x to denote F i x / j and r e c a l l C o r o l l a r y 1.4 in t h i s c o n n e c t i o n . 3 8 V / J a i s a s o l i d t o rus D 2 x S 1 w i th F i x an annulus R e ( 3 D 2 ) x S 1 . V / J M ' i s a s o l i d K l e i n b o t t l e D 2 + X I / - K X ( T | 3 1 ) w i th F i x the Mobius band R e x l . V / J 2 D i s a 3 - c e l l w i th F i x two 2 - c e l l s . V / J D P ' i s C,P wi th F i x the p o i n t v , and a 2 - c e l l . V / J s i s a s o l i d t o rus D 2 x S 1 w i th F i x the 1-sphere OxS 1 . V / J 2 C i s a 3 - c e l l w i th F i x two proper unknot ted 1 - c e l l s . V / J 2 P i s a boundary connected sum of C,P and C 2 P w i th F i x = v i u v 2 ( i . e . ) V / J 2 P = D 2 x S + / ( ( z , 1 ) ~ ( - z , 1 ) , ( z , - 1 ) ~ ( - z , - 1)) where S 4 = D + n S , v , = ( 0 , 1 ) , v 2 = ( 0 , - l ) and D 2 x i i s the connected sum d i s c . V / J N i s a s o l i d K l e i n b o t t l e D 2 X I / K X ( r | 3 1 ) . V / j Q i s a s o l i d t o r u s . C o r o l l a r y 4.3 I f i i s an i n v o l u t i o n on a s o l i d t o rus V then V / t i s i somorph ic to one of the spaces V / j above . The isomorphism type of the f i x e d set and o r i e n t a b i l i t y type of i determine V / t up to i somorph ism. 39 Example 4.4 j Q i s not conjugation extendable because s(z,w)=(z,zw) determines a conjugation s:3V >3V for j 0|3V that does not extend to V. Corollary 4.5 The orientation reversing involutions are conjugation extendable. If V and V are s o l i d t o r i with conjugate orientation preserving involutions t and i' respectively then the involutions are conjugation extendable with respect to the class of isomorphisms 3V > 3 V that: 1 ) extend to isomorphisms V >V , for the case t conjugate to j Q or j<,. 2) extend to isomorphisms V u F i x > V ' u F i x ' , for the case i conjugate to J 2 c Proof: It su f f i c e s to show conjugation extendable for the standard involutions j only. In fact i t s u f f i c e s to show given h ' : 3 V >3V an isomorphism with h'.j.(h' 1 ) = j , that h' extends to an isomorphism H':V >V with H'.j.(H' 1 ) = j . Now h' induces an isomorphism h:3V/j >3V/j. We show, for each j , h extends to an isomorphism H:V/j >V/j. Let p:V >V/j be induced by inclusion. p|(V-Fix) i s a double cover and p|Fix i s an isomorphism. Check that H.(p|): V-Fix >(V-Fix)/j l i f t s to H':V-Fix >V-Fix and thus o b t a i n a c o n j u g a t i o n . Le t W=V/j. When F i x ±0 n o t i c e that h i s on l y d e f i n e d on a proper subman i fo ld (3V)/j of 9 ( V / j ) . The ex t ens i ons H of h are c l e a r f o r J A , 3 2 D ' -'s' -'o' J D P and J M . For J n and J 2 P , ^ w i-s a K l e i n b o t t l e . For J N , the boundary of an e s s e n t i a l proper d i s c D r e p r e s e n t s the unique element of order two in H , (9W)=Z 2 ©Z. S ince h(9D) i s a l s o of o rder 2, h(9D) bounds an e s s e n t i a l p roper d i s c D ' . Extend by con ing over D and then over the 3 - c e l l W-(3WyD). For J 2 p ' 9D 2 x i i s a 1-sphere tha t separa tes 9W and i s 2-s ided in 9W. Such 1-spheres r ep resen t the element (0 ,2 )eH, (9W) . Proceed as above . For j 2 c , l e t h : 9 W u F i x > 9 W ' u F i x ' be g i v e n . A 1-sphere in 9W-Fix tha t decomposes 9 W u F i x i n t o two components each c o n t a i n i n g one component of F i x has the p r o p e r t y that i t bounds a proper 2 - c e l l D in W that misses F i x . Use t h i s 2 - c e l l to extend h. QED Remark 4.6 Cons ide r the s o l i d K l e i n b o t t l e V=D 2 xI/d where d=/cx( T | 91) . Then J a = /cxid/d wi th f i x e d set an annu lu s , J M = -/cxid/d wi th f i x e d set a Mobius band, J D ( , = i d x r / d w i th f i x e d set a 2 - c e l l and a 1 - c e l l , j c p = - K x r / d w i th f i x e d set a p o i n t and a 1 - c e l l , and jg = ax id/d wi th f i x e d set a 1-sphere are the on ly f i v e i n v o l u t i o n s on a s o l i d K l e i n b o t t l e , up to conjugacy. The proof is very s i m i l a r to the one given for the s o l i d torus . Since d i s o r i e n t a t i o n revers ing , however, case 3) does not a r i s e and in case 4) only the combinations that were disal lowed previously can occur. 42 I I I . EQUIVARIANT ANNULUS AND TORUS THEOREMS #5. Annulus Theorems D e f i n i t i o n 5.1 A proper annulus A in a 3-mani fo ld M i s t r i v i a l , i f A decomposes M i n t o a s o l i d t o rus V=D 2 xS 1 and a subman i fo ld M 0 such t h a t : M = M 0 U V , M o n v = 3M0 n 9 V = A and there e x i s t s a nonsepa ra t i ng proper 2 - c e l l B c V w i th B n A = 3 B n A a nonsepa ra t i ng 1-ce l l in A . Otherwise c a l l A n o n t r i v i a l . C a l l V a s o l i d t o rus that t r i v i a l i z e s A. Note tha t i f A does not separa te M or i f 3A i s in d i f f e r e n t boundary components of M then A i s n o n t r i v i a l . D e f i n i t i o n 5.2 C a l l a n o n t r i v i a l i n compres s i b l e proper annulus an e s s e n t i a l annu lus . C a l l an i n c o m p r e s s i b l e proper Mobius band an e s s e n t i a l Mobius band. Let F be a su r f a ce and S a component of F n i F . In some s u r g e r i e s performed l a t e r we w i l l wish to r ep l a ce F by F'=Fx1 where F X [ - 1 , 1 ] i s a b i c o l l a r of F . To i n su re that F * , i F ' and F i x are t r a n s v e r s a l at l e a s t in . s t andard ne ighborhoods (see D e f i n i t i o n s 2 .9- . 10), the f o l l o w i n g lemma 43 i s u sed . Lemma 5.3 Let V be a s o l i d to rus and i : V >V an i n v o l u t i o n . Le t A 0 and A, be a n n u l i in 3V w i th 3A 0 = S 0 | _ | s i ' 3 A 1 = S , u S 2 r A 0 n A i = s i ' i S 0 = S 0 , (A 0 u A, ) n t (A 0 L J A , )=S 0 and F i x n (A 0 u A , ) c S 0 . Then there i s a proper annulus A c V such that A, IA and F i x i n t e r s e c t t r a n s v e r s a l l y w i th A n i A = S a 1-sphere and 3A hav ing one component in i n t ( i A 0 ) and the other component i s S 2 . A s i m i l a r statement ho lds i f V i s a 3 - c e l l and A 0 , A, a re 2 - c e l l s and are 1 - c e l l s . See F i g u r e 6. P r o o f : By t r a n s v e r s a l i t y of F i x , by t a k i n g a s u f f i c i e n t l y sma l l r e g u l a r ne ighborhood N of A 0 u A , u c A 0 u tA, we may assume one of the f o l l o w i n g h o l d s : 1) F i x n N = 0 2) F i x n N c o n s i s t s of two d i s j o i n t 1-ce l l s I. w i th 44 e x a c t l y one p o i n t of 31^ in S 0 and the o ther in i n t ( V ) n 3 N or 3) F i x n N i s an annulus such tha t one boundary component i s S 0 and the o ther in i n t ( V ) n 9 N . F u r t h e r , there i s a r e g u l a r ne ighborhood N' of S 0 c N such tha t iN ' =N' , A 0 ' = N ' n A 0 i s an annu lus , A 0 1 u i A 0 ' = N ' n 3 V and p r o p e r t i e s 1)~3) h o l d w i th r e spec t to N ' . Let B be the annulus which i s the c l o s u r e of N' -A 0 U i A 0 . Then there i s an S in i n t ( B ) w i th tS=S and F i x n B c S. There i s an annulus A" in V-N' such tha t 3A" = S 2 U S and A " n i A " = S . Let A' be the component of B-S tha t meets i A 0 . Then A = A ' U A " i s the d e s i r e d - a n n u l u s . QED Remark 5.4 A s o l i d K l e i n b o t t l e i s a t w i s t e d I-bundle over an a n n u l u s . The annulus i s e s s e n t i a l but i t does not separa te the boundary . Lemma 5.5 If U i s a s o l i d t o rus then U has no e s s e n t i a l a n n u l i . I f U i s a s o l i d K l e i n b o t t l e then U has no e s s e n t i a l a n n u l i tha t separa te 3U. Moreover , suppose A' i s an annulus c o n t a i n e d in 3U such tha t a nonsepa ra t i ng proper d i s c D of U i n t e r s e c t s A ' i n e x a c t l y one nonsepa ra t i ng 1 - c e l l of A ' . I f A i s an 45 i n c o m p r e s s i b l e proper annulus d i s j o i n t from A' then the s o l i d t o rus which t r i v i a l i z e s A may be taken to be d i s j o i n t from A ' . P r o o f : Suppose A i s an e s s e n t i a l annu lu s . Then l e t D be any proper nonsepa ra t i ng 2 - c e l l of U. (When A' i s g i v e n , take D as in the s ta tement . ) Make A and D t r a n s v e r s a l . S ince A i s i n c o m p r e s s i b l e ad jus t D so tha t A n D c o n s i s t s of 1-ce l l s o n l y . If A (-)D=0 then A i s c o n t a i n e d i n a 3 - c e l l ob t a ined by removing a s u f f i c i e n t l y sma l l r e g u l a r ne ighborhood of D from U. T h i s c o n t r a d i c t s i n c o m p r e s s i b i l i t y . If A n D * 0 l e t B be an outermost 2 - c e l l of D (and d i s j o i n t from A' i f A' i s g i v e n ) : so B n A = 3 B n A = I i s a 1-ce l l and B n 3 U = 3 B - I . I f I bounds a 2 - c e l l in A, then by an i so topy moving B, o b t a i n a d i s c D' wi th fewer 1-ce l l s in A n D ' . Assume now that I does not bound a 2 - c e l l in A . Then I sepa ra tes A. Let V be the c l o s u r e of the component of U-A tha t meets i n t ( B ) . 3A decomposes 3U i n t o two a n n u l i or p o s s i b l y , in the case where U i s a s o l i d K l e i n b o t t l e , i n t o an annulus and two Mobius bands. However, in the l a t t e r case 3 B n 3 U must meet the annu lu s . I t f o l l o w s that 3 V n 3 U i s an annulus and V i s a s o l i d t o rus w i th the p r o p e r t i e s making A t r i v i a l . QED 46 We next s t a t e the p a r t i a l annulus theorem and the annulus theorem. The p r o o f s are o m i t t e d . They are s i m i l a r i n s p i r i t to the proof of the to rus theorem. P a r t i a l Annulus Theorem 5.6 Let M be an i r r e d u c i b l e 3-mani fo ld w i th i n v o l u t i o n i . Let A 0 be an e s s e n t i a l annulus w i th 3A 0 r-| 13AO=0. Then : 1) the re i s an e s s e n t i a l annulus A wi th A n i A = 0 and 3ALJ I 3 A = 3 A 0 1 J i 3A 0 or 2) the re are two d i s j o i n t e s s e n t i a l a n n u l i A , , A 2 w i th i A , = A l f i A 2 = A 2 , and 3 ( A , u A 2 ) = 3 A 0 u i 3 A 0 and F i x i s t r a n s v e r s a l to A, and A 2 . Example 5.7 The i n v o l u t i o n Jcxid on D 2 x I induces an i n v o l u t i o n t on RP 2 xI=D 2 xI/d where d=axid i s an i d e n t i f i c a t i o n d e f i n e d on 3D 2 x I=S 1 x I . F i x = ( R e u { i } ) x i . No e s s e n t i a l annulus or Mobius band s a t i s f i e s A n i A = 0 or iA=A and A and F i x t r a n s v e r s a l . There i s an annu lus , however, w i th iA=A but i t i s not t r a n s v e r s a l to F i x . 47 Example 5.8 Cons ide r the n o n o r i e n t a b l e t w i s t e d I-bundle I x l x l / d over a t o r u s , where d = ( T | 9 1 ) x i d x r u i d x ( T | 3 1 ) X T . The i n v o l u t i o n t = i d x T x r / d has Mobius bands but no a n n u l i A w i th Ar-|iA=0 or iA=A and A and F i x t r a n s v e r s a l . Annulus Theorem 5.9 Let M be an i r r e d u c i b l e 3-mani fo ld wi th i n v o l u t i o n i . Suppose A 0 i s an e s s e n t i a l annu lus or Mobius band in M wi th 3 A 0 = S 0 1 u S o 2 where, i f A 0 i s a Mobius band, S 0 1 = S 0 2 « Let R 1 f r e s p e c t i v e l y R 2 , be the component of 3M wi th S 0 i r_ R 1 f r e s p e c t i v e l y S 0 2 c ^2- Assume R, i s i n c o m p r e s s i b l e , and i f R,*R 2 assume a l s o tha t R, i s not a p r o j e c t i v e space . Then the re i s an e s s e n t i a l annulus or Mobius band A wi th e i t h e r A n i A = 0 , or iA=A and A and F i x t r a n s v e r s a l and in both cases 9 A u t 9 A c R l u R 2 | _ , i R 1 | _ | C R 2 . I f M i s o r i e n t a b l e A may be taken to be an annu lus . 48 #6. E q u i v a r i a n t Torus Theorem  Lemma 6.1 Let M be an i r r e d u c i b l e 3-manifo ld c o n t a i n i n g an i n c o m p r e s s i b l e t o r u s . Le t F be a 1-sided K l e i n b o t t l e in the i n t e r i o r of M and W a r e g u l a r ne ighborhood of F in M wi th 3W a t o r u s . Then 9W i s an i n c o m p r e s s i b l e t o r u s . P r o o f : I f not then M=W U U wi th U a s o l i d t o r u s . N e c e s s a r i l y W i s an o r i e n t a b l e t w i s t e d I-bundle over T and M. i s o r i e n t a b l e . The i n c l u s i o n of U in M determines an index two subgroup of ^ ( M ) . Cons ide r p : H >M, the 2-sheeted c o v e r i n g c o r r e s p o n d i n g to tha t subgroup. Then p 1 (W)=Tx[-1,1] where T i s a t o r u s wi th p(TxO)=F. p 1 ( U ) = V 1 U V 2 i s two d i s j o i n t s o l i d t o r i . R i s a l ens space . But M and hence H c o n t a i n s a 2-s ided i n c o m p r e s s i b l e t o r u s . QED E q u i v a r i a n t Torus Theorem 6.2 Let M be an i r r e d u c i b l e 3-mani fo ld wi th i n v o l u t i o n i . Suppose M c o n t a i n s an i n c o m p r e s s i b l e t o r u s . Then one of the f o l l o w i n g h o l d s : (I) There i s a 2-s ided i n c o m p r e s s i b l e t o rus or K l e i n b o t t l e T in int (M) t r a n s v e r s a l to F i x wi th T n i T = 0 or iT=T. 49 ( I I ) M=V. , u V , U U . ! L J U , where and u\ are s o l i d t o r i and t V i = V i and t U . , = U , . There are a n n u l i A^, i = ± 1 , w i th A i n A . , = A i n i A i = 9 A i = 3 i A . = V , n V . , = U , n U . , and V - n U . ^ A j , V . n U _ . = i A i f a v ^ A - L j i A . , au .-Aj u tA_. . See F i g u r e 7. A , | _ | A . 1 i s a 2-s ided i n c o m p r e s s i b l e t o rus or K l e i n b o t t l e t r a n s v e r s a l to F i x . c jV^ i s o r i e n t a t i o n p r e s e r v i n g . ( I I I ) M = V 1 U V 2 U V where V , , V 2 and V are s o l i d t o r i each i n v a r i a n t under i such that t i s o r i e n t a t i o n p r e s e r v i n g when r e s t r i c t e d to any of V , , V 2 and V. There i s a 1-sided K l e i n b o t t l e T w i th T n i T = S £ i n t (V ) a gene ra to r of 7 r , (V ) . V , n V 2 = ( T n tT) - i n t (V ) are two a n n u l i . T , iT and F i x are p a i r w i s e t r a n s v e r s a l and F i x n 3 V 2 = 0 and F i x n S * 0 . V i s a s t andard ne ighborhood of S. See F i g u r e 7. (IV) M=W U V where W i s a t w i s t e d I-bundle over a t o rus T £ W and V i s a s o l i d t o rus w i th 3W=3V=W nV and tW=W, iT=T and tV=V. F i x i s t r a n s v e r s a l to 3W and T except fo r a p o s s i b l e 1-sphere component S of F i x 1 c o n t a i n e d in T . P r o o f : Let T 0 be an i n c o m p r e s s i b l e t o rus in i n t ( M ) . By Theorem 2 .12 assume T 0 , t T 0 and F i x are a lmost p a i r w i s e t r a n s v e r s a l and tha t no 1-spheres in T 0 n i T 0 bound 2 - c e l l s in T 0 . ( I I ) ( I I I ) -1 F i g u r e 7. As a f i r s t s t ep we handle the cases where sadd le components a r i s e . Only Type III and Type II components are p o s s i b l e . In both cases s i n ce S and i n t e r s e c t t r a n s v e r s a l l y at one p o i n t , the re can be on l y one component in T 0 n t T o • Suppose T o n i T 0 i s a Type III component S U S 1 U S 2 . Then S, and S 2 bound an annulus A in T 0 s i n ce S i n S 2 = 0 and both i n t e r s e c t S t r a n s v e r s a l l y once . Let T = A u i A . Then t,T=T and T and F i x are t r a n s v e r s a l . T i s 1-sided s i n c e a r e g u l a r ne ighborhood of S, i s a s o l i d K l e i n b o t t l e . Let N be a r e g u l a r ne ighborhood of T i n v a r i a n t under N . I f 9N i s i n c o m p r e s s i b l e then i t i s a 2-s ided to rus s a t i s f y i n g ( I ) . If 3N i s compress i b l e we a r r i v e at ( IV ) . Suppose T 0 n i T 0 i s a Type II component S u S , . F i r s t we c o n s t r u c t a t o rus T ' i s o t o p i c to T=T 0 w i th i T ' = T ' . Let N ( S ) and N(S, ) be r e g u l a r ne ighborhoods of S and S, r e s p e c t i v e l y , both i n v a r i a n t under i such tha t N = N (S) LJ N (S, ) i s a r e g u l a r ne ighborhood of S U S , and such tha t T N N ( S ) and T n N ( S , ) a re 51 a n n u l i , N ( S ) n F i x i s a Mobius band, N ( S 1 ) n F i x i s a proper 2 - c e l l and N ( S , ) n F i x 1 i s a p roper 1 - c e l l . Both N(S) and N fS , ) are K l e i n b o t t l e s . By t r a n s v e r s a l i t y the re are two d i s j o i n t open 2 - c e l l components K, and K 2 of N ( S ) - ( T u i T ) tha t meet F i x 1 and the re are two d i s j o i n t open 2 - c e l l 2 components L, and L 2 of N ( S , ) - ( T U tT) that do not meet Fix . By c o n s i d e r i n g the e f f e c t of t near sadd le p o i n t s we see A = ( K , u K 2 u L , u L 2 ) n 3 N i s an annulus wi th 3 A = C u i C where C = 3 N n T . The c l o s u r e of A u (T-N) u i (T-N) i s a 2-sphere which by the i r r e d u c i b i l i t y of M bounds a 3 - c e l l E. E cannot c o n t a i n the proper punc tured to rus T n N so E n in t (N ) =0. S ince F i x 1 i s t r a n s v e r s a l to 3E and i3E=3E i t f o l l o w s iE=E. In p a r t i c u l a r t|E i s con jugate to j l r the s tandard i n v o l u t i o n of a 3-ce l l w i th f i x e d set one 1 - c e l l . A i s i n v a r i a n t and c o n t a i n s F i x 1 n 3 E . Hence one shows the re i s a proper 2 - c e l l D w i th 3D a genera to r of H,(A) such that F i x 1 n E i s a proper 1-ce l l of D and iD=D. S ince i3D=3D, by t a k i n g N s u f f i c i e n t l y sma l l we can c o n s t r u c t a proper punc tured t o r u s P in N w i th 3P=D and iP=P (namely i so tope 2 T n N ) . Cons ide r the t o rus T * = P U D . Fix i n t e r s e c t s T ' t r a n s v e r s a l l y at S, and F i x 1 i s c o n t a i n e d in T ' . T ' i s 1-s ided. Le t W be a r e g u l a r ne ighborhood of T* i n v a r i a n t under i . I f 3W i s i n c o m p r e s s i b l e then i t i s a 2-s ided t o rus s a t i s f y i n g ( I ) . I f 3W i s compres s i b l e we a r r i v e at case ( IV ) . 52 We may now assume T 0 n t T o has no sadd le components. T 0 , i T 0 and F i x are p a i r w i s e t r a n s v e r s a l and T 0 n i T 0 c o n s i s t s of d i s j o i n t 1-spheres bounding a n n u l i i n T 0 and i T 0 . We  s u c c e s s i v e l y c o n s t r u c t i n c o m p r e s s i b l e t o r i or K l e i n b o t t l e s  T wi th fewer 1-spheres in T n i T , but a lways keep T n i T c o n s i s t i n g of 1-spheres bounding a n n u l i in T and i T . T h e r e f o r e any 1-sphere of T n i T w i l l a lways have a s tandard ne ighborhood . See D e f i n i t i o n s 2 . 9 - . 1 0 . It a l s o f o l l o w s then that any 1-sided K l e i n b o t t l e a r i s i n g from such " a c o n s t r u c t i o n has a r e g u l a r ne ighborhood W w i th 3W a t o r u s . So Lemma 6.1 i s a p p l i c a b l e . No te : Suppose T s a t i s f i e s a l l the c o n d i t i o n s of (I) except tha t T i s 1-sided i n s t e a d of 2-s i ded . Le t W be a r e g u l a r ne ighborhood of T . We can take W so tha t 3W and F i x are t r a n s v e r s a l and tW=W or W niW=0. 3W i s 2 - s i d e d . I f 3W i s i n c o m p r e s s i b l e , 3W s a t i s f i e s ( I ) . I f 3W i s c o m p r e s s i b l e , by Lemma 6.1 T i s a t o r u s . Now V=M-W i s a s o l i d t o r u s . I f iT=T we have ( IV ) . I f i T n T=0 then the s o l i d t o r u s V c o n t a i n s an embedded 1-sided to rus i T , a c o n t r a d i c t i o n . There are four main cases now depending on the number of 1-spheres of T n i T and the c o m p r e s s i b i l i t y of c e r t a i n sur f a c e s . Assume T n i T c o n s i s t s of at l e a s t two 1-spheres. Let A £ iT be an innermost a n n u l u s : Ar-|T=3A. 3A decomposes T i n t o two a n n u l i A 1 and A " w i th T = A ' , , A n and F i g u r e 8. 3A=3A'=3A"=A' n A " . T ' = A ' U A and T " = A " U A are t o r i or K l e i n b o t t l e s . See F i gu re 8. Case 1) T ' i s i n c o m p r e s s i b l e . Case 1 . 1 ) t3A=3A and iA=A ' . Then c T ' = T ' . One sees Fix i s t r a n s v e r s a l to T ' by c o n s i d e r i n g the s tandard ne ighborhoods of 3A. We a r r i v e a t case (I) or ( IV ) . Case 1 . 2 ) E i t h e r i3A=3A and iA=A" or i 3 A n 3 A i s a s i n g l e 1-sphere S and IA £ A " . In the l a t t e r case tS=S. Let V, and V 2 be d i s t i n c t s t anda rd ne ighborhoods of 3A and l e t 7, and y2 be the two d i s t i n c t s t andard a n n u l i tha t meet both A and A ' . Let T , (A' U A U 7 , U 7 2 ) " i n t ( V 1 u V 2 ) . Then T ^ i T , L ( T n i T ) -3A because ( 7 i [ _ j 7 2 ) n 1 (71u72 ) = 0 « T ' and T , are ambient i s o t o p i c so T , i s i n c o m p r e s s i b l e . F i x n ( 7 , u 7 2 ) = 0 and A i s innermost so T , , iT^ and Fix are p a i r w i s e t r a n s v e r s a l . P roceed w i th T , . 54 Case 1.3) E i t h e r t 9 A n 9 A i s a s i n g l e 1-sphere S and I A £ A ' or i9A n9A=0. Let 3 A = S U S ' . Let V be a s t andard ne ighborhood of S and l e t 7 be the s tandard annulus tha t meets both A and A ' . Le t A x [ 0 , e ] be a s u f f i c i e n t l y t h i n c o l l a r of A=AxO in M such tha t S ' x [ 0 , e ] C A ' , S x [ 0 , e ] r - T a n d ( A n 3 V ) x [ 0 , e ] = (Ax[0,c3) n 3 V . The c o l l a r e x i s t s s i n ce V i s a s o l i d t o r u s . In the f i r s t c a s e , tS = S and 1 7 = 7 . By Lemma 5.3 i f (Axe) r-| 7*0 we may assume ( A x e ) n V and t ( ( A x e ) n V ) i n t e r s e c t t r a n s v e r s a l l y in a 1-sphere S, and tha t both are t r a n s v e r s a l to F i x . In a 1-1 o ther cases set S,=S. De f i ne T , = (Axe) u A ' - ( ( S ' u S ) x L 0 , e j ) u ( S x [ 0 , e ] ) - A ' Then T i n i T , L ( ( T n t T ) - 9 A ) u S , . T , i s i n c o m p r e s s i b l e ( s i n c e i t i s ambient i s o t o p i c to T ' . T 1 f i T , and F i x are p a i r w i s e t r a n s v e r s a l . P roceed wi th T , . By case l ) we may now assume T ' and T " are c o m p r e s s i b l e . Case 2) For every annulus A £ i T w i th A n T = 3 A , both c o r r e s p o n d i n g su r f a ce s T ' and T " are c o m p r e s s i b l e and T n t T c o n t a i n s more than two 1-spheres. Then l e t A, and A 2 in i T be a n n u l i w i th A ^ n T = 9 A ^ and w i th 9A^ = S 0 L j S i w h e r e S ' s ' a n < ^ s 2 a r e 1-spheres w i th 55 S , * S 2 . Let A, A,' and A 2' be the th ree a n n u l i of T tha t these 1-spheres decompose T i n t o : 9A=S1LJS2 and 9 A ^ ' = S 0 u S ^ , i = 1 , 2 . See F i g u r e 9. De f i ne T , =A u A , u A 2 . T , i s i n c o m p r e s s i b l e . Otherwise T , bounds a s o l i d t o rus or a K l e i n b o t t l e U. Say A,' c U. A,' i s t r i v i a l in U by Lemma 5 .5 . If A , ' u A , bounds the t r i v i a l i z i n g to rus then the i n c o m p r e s s i b l e T=A , ' u A 2 ' |_, A i s ambient i s o t o p i c to A 1 U A 2 ' U A which was compres s i b l e by h y p o t h e s i s . If A / L J A ^ L J A bounds the t r i v i a l i z i n g t o r u s then , s i n c e A,' and A 2 meet on S 0 , A 2 must a l s o be t r i v i a l in A 1 ' u A 2 | _ j A . So T i s ambient i s o t o p i c to A 2'|_jA 2 which was assumed c o m p r e s s i b l e . We have f i v e c a s e s : Case 2.1) i (S, u S 2 ) = S , u S 2 and i S 0 r. A. Then i ( A 1 u A 2 ) = A . Case 2.2) i (S , u S 2 ) =S , u S 2 and cS 0 r j A ' . Then t ( A , u A 2 ) = A , ' u A 2 ' and t S 0 = S 0 . Case 2.3) t S ^ S , and t S 0 = S 0 . Then tA,=A, ' and t S 2 c A 2'. Case 2.4) i ( S , u S 2 ) n ( S , u S 2 ) = 0. 56 Case 2.5) i ( S , y S 2 ) n (Si U S 2 ) i s a one 1-sphere. These cases cover a l l p o s s i b i l i t i e s . In each case we f i n d a T^ w i th fewer 1-spheres. In case 2.5) t h i s f o l l o w s from the o the r c a s e s . A f t e r r e l a b e l l i n g assume S, i s the 1-sphere in the i n t e r s e c t i o n . Then t S ^ S , . By case 2.3) we assume tSo^So. Let A 3 be the innermost annulus ad jacen t to A , : A 3 c tT , w i th A 3 n T , = 3 A 3 = S , | j S 3 where S 3 * S 0 . By case 2.3) aga in we may assume t S 3 * S 3 . By case 2.1) and 2.2) we may assume t S 0 * S 3 . So we have i ( S 0 u s 3 ) n ( s o u s 3 ) = 0 a n ^ c a s e 2.4) g i v e s the reduct i o n . In case 2.1) use T ^ t T , . F i x i s t r a n s v e r s a l to T , s i n ce the s tandard annulus meet ing A, and A i s i n v a r i a n t . In case 2.2) and case 2 . 3 ) : For i=1,2 l e t be the s tandard ne ighborhoods of S^  w i th 7^ the s t anda rd a n n u l i tha t meet both A and A^. In a l l cases ( 7 i | _ j 7 2 ) n u 7 2 ) = 0. D e f i n e T 2 to be the i n c o m p r e s s i b l e s u r f a c e ambient i s o t o p i c to T , g i ven by T 2 = T 1 U 7 1 U 7 2 - i n t ( V 1 u V 2 ) . Then T 2 , t T 2 and F i x are p a i r w i s e t r a n s v e r s a l and T2 n '^2 c T , j—j tT 1 - (S , |_ jS 2 ) . In case 2 . 4 ) : F i r s t assume t S 0 * S 0 . By symmetry assume tSo^S , . Let ( A ! u A 2 ) x [ 0 , e ] be a s u f f i c i e n t l y t h i n c o l l a r of A i u A 2 = ( A i u A i ^ x 0 i n M s u c h t h a t (S , u S 2 ) x [ 0 , e ] c T and S o x [ 0 , e ] c A , . De f i ne T 2 as 57 ( A , u A 2 ) x e u A - ( ( S , U S 2 ) x L 0 , e J ) u ( ( S , u S 2 ) x [ 0 , e ] - A ) . Then T 2 n i T 2 ^ i ( T n i T ) - S 0 . T 2 i s ambient i s o t o p i c to i n c o m p r e s s i b l e T , . T 2 , t T 2 and F i x are p a i r w i s e t r a n s v e r s a l . If i S 0 = S 0 , p roceed as above but r ep l a ce the c o n d i t i o n S o x [ 0 , c ] £ A, by S , x [ 0 , e ] £ A. Use Lemma 5.3 on a s t anda rd ne ighborhood of S 0 to ad ju s t the c o l l a r so that ( A 1 u A 2 ) x e and t ( A , u A 2 ) x e i n t e r s e c t t r a n s v e r s a l l y in one 1-sphere S 3 . Then T 2 n i T 2 c ( i ( T n t T ) - ( S 0 L J S , ) ) u S 3 . Case 3) For each annulus A c iT wi th A n T = 3 A , both c o r r e s p o n d i n g s u r f a c e s T ' and T " are-not i n c o m p r e s s i b l e and T n t T i s e x a c t l y two 1-spheres. Set t T = A . 1 u A 1 w i th A . , n A , = 3 A , = 3 A . , = T n i T = S , u S 2 . Then T= IA. , U IA , . There are s o l i d t o r i or K l e i n b o t t l e s U. and V l ( i = ± l ) p a i r w i s e d i s j o i n t on t h e i r i n t e r i o r s wi th 3 V ^ = A ^ u t A ^ and 3U^=A^ |_j tA_^. None of U^ or are s o l i d K l e i n b o t t l e s . O the rw i se , i f say V, i s a s o l i d K l e i n b o t t l e , then s i n c e S, decomposes 3V, i n t o two a n n u l i i t f o l l o w s tha t S, bounds a d i s c in V , . T h i s c o n t r a d i c t s the i n c o m p r e s s i b i l i t y of T . By c o n s i d e r i n g the s tandard a n n u l i of a s tandard ne ighborhood of S, we see i V ^ V ^ and I U ^ U ^ . Next we show i |V, and t | V . , are o r i e n t a t i o n p r e s e r v i n g . I f not then by S e c t i o n 4, 1 1 i s con juga te to J A , J 2 r j f J N f J m or J D P » the s tandard i n v o l u t i o n s on a s o l i d t o r u s . and J D P are not p o s s i b l e s i n ce S, or S 2 would bound a d i s c 58 c o n t r a d i c t i n g the i n c o m p r e s s i b i l i t y of T . I f 11V-, i s con juga te to J M then say S, £ F i x and S 2|-)Fix=0. Then t |V 2 has a 2-d imens iona l f i x e d set component tha t has on l y one boundary component. It f o l l o w s t |V 2 i s a l s o con juga te to J M > So F i x c o n t a i n s a K l e i n b o t t l e K. There i s a r egu l a r ne ighborhood W of K w i th t9W=9W and W n F i x = 0 . S ince V\ are s o l i d t o r i and K n V \ i s a Mobius band, 3W i s a t o r u s . By Lemma 6 . 1 , 3W i s i n c o m p r e s s i b l e . We a r r i v e at ( I ) . If t |V, i s con juga te to J A , then [S , ] r ep r e sen t s a genera to r of H, (V , ) and hence the re i s an ambient i so topy t a k i n g tT to 9U. , (move A, to i A , ) . T h i s c o n t r a d i c t s tha t tT i s i n c o m p r e s s i b l e . F i n a l l y suppose t |V , i s con jugate to the i n v o l u t i o n J N = K x a on D 2 xS 1 . I f tS,=S, then S , ' = 1 xS 1 and S j ^ - l x S 1 determine a n n u l i A' and iA ' of 9D 2xS 1 . I t i s p o s s i b l e to c o n s t r u c t a c o n j u g a t i o n 9V, >9D 2 xS 1 t a k i n g A, to A ' . T h i s c o n j u g a t i o n extends to a c o n j u g a t i o n V, > D 2 x S ' . But [ S , ' ] i s a genera to r of H , ( D 2 x S 1 ) and we get a c o n t r a d i c t i o n as f o r the J A case above. I f t S 1 =S 2 then use S,'=9D2x1 and S 2'=9D 2x-1 and proceed as above but t h i s t ime o b t a i n i n g a c o n t r a d i c t i o n as fo r j 9 n above. 59 Case 4) T n i T i s a s i n g l e 1-sphere S. Then tS=S. Let V be a s t andard ne ighborhood of S and l e t a , , a2, 01 and 0 2 be t^ e s t andard a n n u l i w i th a i n a 2 = 0 , /3 i n/3 2 =0, i a , = a , , t a 2 = a 2 and t j3 •, = /? 2 . De f i ne T , = ( T u t T u a , u a 2 ) - i n t (V ) and T 2 = ( T u i T u 0 , u 0 2 ) - i n t ( V ) . If T i s 2-sided then T , i s 2-s ided . A l s o t T ^ T , . S ince T i s 2-s ided i t f o l l o w s tha t a s u f f i c i e n t l y t h i n c o l l a r T x [ 0 , e ] of T=TxO can i n t e r s e c t on ly one of i n M c ^ ) and i n t ( a 2 ) . Hence T , cannot separa te and t h e r e f o r e T , i s i n c o m p r e s s i b l e . We a r r i v e at ( I ) . From now on assume T i s 1 - s i d e d . T , and T 2 are t o r i . T h i s f o l l o w s s i n c e V i s a s o l i d t o rus and e i t h e r both of the a n n u l i T - i n t ( V ) and tT- in t (V ) a re " t w i s t e d " r e l a t i v e to V ( i f T i s a K l e i n b o t t l e ) or n e i t h e r i s ( i f T i s a t o r u s ) . If e i t h e r of T , or T 2 i s i n c o m p r e s s i b l e we a r r i v e at ( I ) . Assume then tha t T , and T 2 are c o m p r e s s i b l e . Then T^ bounds a s o l i d to rus V^. I f S c V\ then c o n t a i n s a 1-s ided t o r u s or K l e i n b o t t l e , a c o n t r a d i c t i o n . So M = V U V 1 U V 2 w i th i n t ( V ) , i n t ( V , ) and i n t ( V 2 ) p a i r w i s e d i s j o i n t . By c h o i c e of a, and a2, t i n t e r changes the components of ba^. T h e r e f o r e t|a^ i s con juga te to one of i d x r , K X T or a x r , the s tandard i n v o l u t i o n s of S ' x I . Let S. be a 1-sphere of a. tha t i s the image of S'xO under some c o n j u g a t i o n . 60 Note tha t does not bound a d i s c D in V , , o the rw ise T would be c o m p r e s s i b l e . Note a l s o that i f the re i s an annulus A [ V , w i th 3 A = S , U S 2 and iA=A then we a r r i v e at p r o p e r t y (I) and (IV) as f o l l o w s . Torus V, i s separa ted by A. S ince i i n t e r changes the components of 3 a , , c i n t e r changes the components of A-V , . A i s t r i v i a l in V, so i t f o l l o w s V, can be g i ven a t r i v i a l I-bundle s t r u c t u r e over A. There i s an annulus B [ V w i th 36=5, | | S 2 and iB=B. V i s an I-bundle over B. Cons ide r T 3 = A U B . It f o l l o w s V U V , i s an I-bundle over T 3 w i th 3 ( V U V , ) = T 2 a t o r u s . Moreover T 3 does not separa te T so T 3 i s 1-s ided. I f T 3 i s a t o rus we a r r i v e at ( IV ) . I f T 3 i s a K l e i n b o t t l e , Lemma 6.1 g i v e s ( I ) . S ince V i s a s tandard ne ighborhood , F i x n a , = 0 i f and on ly i f F i x n a 2 = 0 . T h e r e f o r e t | a , and i | a 2 a re c o n j u g a t e . Case 4.1) t | a 1 i s con juga te to idxT. Then t |V , has a 2-d imens iona l f i x e d set tha t meets 3V, in two f i x e d 1-spheres. By S e c t i o n 4 i t f o l l o w s tha t S, bounds a d i s c or S 1 U S 2 bound an annulus A f i x e d by 11V, . By the above comments, we a r r i v e at (I) or ( IV ) . Case 4.2) i | a , i s con juga te to axr. Then i |V, i s o r i e n t a t i o n r e v e r s i n g , i | 3V, i s con juga te to ax/c on S 1 x S 1 by a c o n j u g a t i o n t a k i n g S^  to S ' x t - I ) 1 . By S e c t i o n 4 t |V, i s con juga te to S.XK or axk by a c o n j u g a t i o n ex tend ing the one g i ven on the b o u n d a r i e s . In the f i r s t case S, bounds a d i s c 61 and in the second case S 1 U S 2 bound an annulus w i th iA=A. Aga in by the above comments we a r r i v e at (I) or ( IV ) . Case 4.3) L\CI-\ i s con juga te to KXT . Then i |V , and t|V are o r i e n t a t i o n p r e s e r v i n g . Now t |V 2 i s o r i e n t a t i o n r e v e r s i n g i f and on ly i f T i s a t o r u s . To see t h i s l e t S^a,!-)/^ and wi thout l o s s say S, £ T . O r i e n t S , . S, and tS, bound two a n n u l i A, and A 2 of 9V 2 w i th i A , = A 2 . Cons ide r the ways of i nduc i ng an o r i e n t a t i o n on i S , . The o r i e n t a t i o n induced by A, and the o r i e n t a t i o n induced by a , a re the same i f and on l y i f T i s a t o r u s . S ince i | a , i s o r i e n t a t i o n r e v e r s i n g the o r i e n t a t i o n induced by a , and the o r i e n t a t i o n induced by i are o p p o s i t e . So i and A, induce oppos i t e o r i e n t a t i o n s on tS, i f and on ly i f T i s a t o r u s . S ince iA ,=A 2 the c l a i m f o l l o w s . If T i s a t o rus then t |V 2 i s o r i e n t a t i o n r e v e r s i n g so i |9V 2 i s con juga te to the i n v o l u t i o n ax/c on S 1 x S 1 by a c o n j u g a t i o n t a k i n g S, to i x S 1 . As in case 4.2) we a r r i v e at (I) or ( IV ) . If T i s a K l e i n b o t t l e then we a r r i v e at ( I I I ) . 119V 2 i s f i x e d p o i n t f r ee so i |V 2 i s con jugate to j g or j ^ wh i le t |V , i s con juga te to J 2 c QED IV. INVOLUTIONS ON ORIENTABLE I-BUNDLES OVER TORI AND KLEIN BOTTLES #7. I n v o l u t i o n s on the T r i v i a l I-Bundle Over a Torus As an a p p l i c a t i o n of the annulus theorem we c l a s s i f y the i n v o l u t i o n s on v a r i o u s I-bund les . D e f i n i t i o n 7.1 Let W=S 1xS 1xI be the t r i v i a l I-bundle over the to rus T = S 1 x S 1 . De f i ne the f o l l o w i n g i n v o l u t i o n s on W (see F i g u r e 10) : k T = i d x i d x T hav ing f i x e d set the t o rus S 1 x S 1 x O k 2 A = i d x K x i d hav ing f i x e d set two a n n u l i S 1 x ± 1 x l k 2 g = i d x K X T hav ing f i x e d set two 1-spheres S 1 x±1x0 k A= (p. ( idx(c) ) x id hav ing f i x e d set the annulus S 1 x 1 x l k g = ( p . ( i d x K ) ) x r hav ing f i x e d set the 1-sphere S 1 x1x0 k 4£=KX/<xid hav ing f i x e d set four 1-ce l l s ±1x±1xl k^p=/cx«XT hav ing f i x e d set four p o i n t s ±1x±1x0 k Q F = a x i d x i d k V T T =axidx7 NI kV7 =ax/<xid NF k Q I = a x « X T Here p . (idxK ) ( z , w ) = ( z w , w ) . The l a s t four i n v o l u t i o n s are f i x e d p o i n t f r e e . The s u b s c r i p t 0 means the i n v o l u t i o n i s o r i e n t a t i o n p r e s e r v i n g , N means i t i s o r i e n t a t i o n r e v e r s i n g , F means i t keeps the boundary components f i x e d (as s e t s ) , k4P (•Ov^»(«C?^ 64 and I means i t i n t e r changes the two boundary components. The o r i e n t a t i o n type of the o ther i n v o l u t i o n s i s determined by the d imens ions of t h e i r f i x e d p o i n t s e t s . These e leven i n v o l u t i o n s are not c o n j u g a t e , because f i x e d p o i n t s e t s , o r i e n t a b i 1 i t y type and F/I. p r o p e r t i e s are con jugacy c l a s s i nva r i a n t s . and kg are con juga te to the f o l l o w i n g . On S 1 x l x l d e f i n e the i d e n t i f i c a t i o n d = a x T | g j X i d . Then W = S 1 x l x l / d . . Let k A ' = ( i d x r x i d ) / d w i th f i x set ( S ' x O x U / d and k s ' = ( i d x r x T ) / d w i th f i x set ( S 1 x O x O ) / d . The other i n v o l u t i o n s can be g i ven s i m i l a r a l t e r n a t e con jugate r e p r e s e n t a t i o n s us ing d = a x r | x i d or d = i d x r | x i d . Lemma 7.2 Let A be an e s s e n t i a l annulus in T X [ - 1 , 1 ] , Then (Tx [-1 ,1 ] ,A ) = ( S 1 X S 1 X [ - 1 , 1 ] , S ' X 1 X [ - 1 , 1 ] ) . P r o o f : By an isomorphism take one of the boundary components of A as S ' x l x l - I , 1 ] . F i r s t show A meets both boundary components of T X [ - 1 , 1 ] . If not then l e t A' be the annulus S 1 X J X [ - 1 , 1 ] where J i s an i n t e r v a l chosen s u f f i c i e n t l y c l o s e to l e S 1 and sma l l enough so tha t A ' n 3 A = 0 and A' i s in the component of T X [ - 1 , 1 ] - A tha t meets T x - 1 . Then, in the s o l i d t o rus T x [ - 1 , 1 ] / ( z , w , - 1 ) ~ ( z , w ' , - 1 ) , the d i s c 1 x S 1 x [ - 1 , 1 ] / ~ i s a 65 nonseparating 2- c e l l meeting A' in one nonseparating 1-cell of C. By Lemma 5.5 there i s a s o l i d torus V that t r i v i a l i z e s A and does not meet A'. Necessarily V does not meet Tx-1 so A i s also t r i v i a l in T X [ - 1 , 1 ] . This i s a contradiction. Next adjust by an isomorphism so that A and 1xS 1x[-1,1] meet in a single proper 1 - c e l l . Then the isomorphism can be constructed. QED Theorem 7.3 Let i and i ' be involutions on W=S1xS1xI with isomorphic fixed point sets. If t and t 1 are fixed point free ' assume, in addition, that both have the same orientation type and that c interchanges boundary components of 3W i f and only i f t' interchanges boundary components of 9W. Then i and t' are conjugate. An involution on W i s conjugate to one of the eleven involutions l i s t e d above. Proof: Let i be an involution on W=S1xS1xI. We show i t i s conjugate to a standard involution. By the Annulus Theorem 5.9 there i s an essential annulus A with either tA nA=0 or iA=A and A and Fix transversal. In the l a t t e r case, by Lemma 3.5 assume the c o l l a r of A i s not interchanged. By the previous lemma take A of form S 1x1x[-1,l]. With further 66 adjustment take iA=S 1 x-1x[- 1,1] i f iA*A . Let W4 = S ' x { x + y i : y > 0 } x [ - l ( l ] and W . = S 1 x { x + y i : y < 0 ) x [ - l j ] . There are three c a s e s : Case 1) iA=A, A and F i x are t r a n s v e r s a l and the c o l l a r of A i s not i n t e r c h a n g e d . Case 2) iA n A = 0 and iW+=W+. Case 3) iA n A = 0 and iW + =W.o. We show that i i s con juga te t o : in case 1) k Q F , * m , k T , k 2 A , k 2 g , k A or k g in case 2) k N p , k Q I , k 2 A , k 2 g , k f t , k g , k 4 C or k 4 p . in case 3) k Q F , k N J , k N p or k Q I . S eve ra l of the s tandard i n v o l u t i o n s are l i s t e d in more than one c a s e . Each s tandard i n v o l u t i o n (or at l e a s t a con jugate of one) can in f a c t a r i s e in the case i t has been l i s t e d under . To see t h i s i t s u f f i c e s to d i s p l a y an annulus A' in W, not n e c e s s a r i l y of form S 1 x 1 x l , w i th p r o p e r t i e s ana logous to those of A. Cons ide r A' as f o l l o w s : in case 3) take I x S ' x I ; in case 2) take S 1 x i x l ; in case 1) take S ' x l x l fo r k Q F ' k N I ' k T ' t a k e 1xS 1 xI f o r k 2 A ' k 2 S a n d t a k e { ( z , z 2 , t ) : z , t } fo r k A , kg . C a l l two i n v o l u t i o n s on W of same type i f they have i somorph ic f i x e d se t s and , in a d d i t i o n , when they are f i x e d p o i n t f r e e , i f they have the same o r i e n t a t i o n type and s i m u l t a n e o u s l y i n te r change or do not in te r change boundary 67 components. It s u f f i c e s to show, f i r s t , tha t t has the same type as a s tandard i n v o l u t i o n l i s t e d under a co r respond ing , c a s e , and second, i f i and i ' have the same type and f a l l i n t o the same case 1) - 3) then they are c o n j u g a t e . Constant use i s made of S e c t i o n 4. Reserve j to denote s t anda rd i n v o l u t i o n s on the s o l i d t o r u s . A l l c o n s t r u c t i o n s done f o r t a re to be per formed fo r t ' , even i f not e x p l i c i t l y s t a t e d . Case 1) iA=A, A and F i x are t r a n s v e r s a l and the c o l l a r of A i s not i n t e r c h a n g e d . Then W=S 1 xlx[- 1,1]/d where d = i d x ( T | 3 1 ) x i d . The i n v o l u t i o n t induces an i n v o l u t i o n X on the s o l i d t o rus V = S 1 x l x [ - 1,1] w i th the p r o p e r t y X.d=d.X when r e s t r i c t e d to S ' x3 Ix [ - 1,1 ]. F i x (X ) i s proper s i n c e F i x i s t r a n s v e r s a l to A . Le t A a l s o denote the copy S 1 X I X 1 in V . S ince the c o l l a r i s not i n te r changed X(A)=A. See Remark 3 .6 . By a d j u s t i n g c i n a c o l l a r of A we may assume t | A and X|A a re one of the f i v e s tandard i n v o l u t i o n s on an annulus (Lemma 3.2). ' Let S be a f i x e d component of 3A. [S]eH,(V)=Z i s a g e n e r a t o r . Wr i te V = D 2 x S 1 . Let A^S ' x l and L=1xS 1 . Then [M] and [L] generate H,(3V)=Z@Z and with a proper cho i c e of o r i e n t a t i o n s [S]=[L]+a[M] where aeZ . X i s not con juga te to j g , J M , J D p or J 2 p : J f ^ were con juga te to j g t hen , s i n ce jg i s o r i e n t a t i o n p r e s e r v i n g and 3F ix ( j ' s ) = 0 , X |A=axid . T h e r e f o r e S i s kept se tw ise f i x e d by 68 X. So [S/X] r ep resen t s twice the genera to r of H 1 (V/X )=Z. However [ F i x ( X ) ] a l s o r ep r e sen t s a genera to r of H,(V) and [ F i x ( X ) / i ] i s a genera tor of H , (V/X). If X were con juga te to j w then t # [ S ] = i * [ L ]+a i»[M] = [L]+[M]-a[M] * ± [ S ] c o n t r a d i c t i n g <.A=A. If X were con jugate to j ^ p then i*[S] = ~[L]-[M]+a[M] t ± [ S ] c o n t r a d i c t i n g tA=A. I f X were con juga te to J 2 P then s i n ce j 2 p i s o r i e n t a t i o n r e v e r s i n g and 9 F i x ( J 2 P ) = 0, X|A=axT. The re fo r e [ S ]= i * [ S ] . T h i s i m p l i e s [L]+a[M] = ~[L]+a[M], a c o n t r a d i c t i o n . Hence X i s con jugate to J A , J 2 D , J 2 C , j Q or J N > Le t B be a component of V - i n t ( A u d ( A ) ) tha t meets S. We i n v e s t i g a t e the f i v e p o s s i b i l i t i e s fo r X |A . S ince we w i l l see these g i ve r i s e to i n v o l u t i o n s of d i f f e r e n t t y p e s , s e l e c t a c o n j u g a t i o n h:A >A' between X|A and X' |A ' and choose S '=h (S ) . T h i s c o n j u g a t i o n extends to a c o n j u g a t i o n h : A u d ( A ) >A' u d ' (A' ) . Case 1 . 1 ) X|A=KXT. Le t F ix (X |A )= {x , y } . N e c e s s a r i l y X i s con jugate to 32C' J ^ x a n d °- ( x ^ a r e * n t ^ i e s a m e component of F i x(X) then F i x i s two 1 -sphe re s . Otherwise F i x i s one 1-sphe re . So i has the type of k g or k 2 s ' N o w x ' a n d d ' ( x ' ) a re in the same component of F i x(X') i f f x and d(x) are in the same component of F i x ( X ) . The c o n j u g a t i o n extends over F i x so extend i t to a c o n j u g a t i o n h:V >V between X and X'. A c o n j u g a t i o n h:W >W' between t and i ' i s i nduced . 69 Case 1.2) X|A=/<xid. Proceed as in case 1.1) except now F ix (X |A ) i s two 2 - c e l l s , so F i x i s e i t h e r two or one a n n u l i . Thus i has type of k 2 A or k A . Now iB=B. h extends over B s i n c e F i x sepa ra tes B i n t o two components. Extend s i m i l a r l y over the annulus 9V-(B u A U IA) . Case 1.3) X |A= idxr . Then X i s con juga te to J A so F i x i s a to rus and t has the type of k T . Proceed as in 1.1). Case 1 . 4 ) X|A=axr. I t f o l l o w s XB n B=0 . So X i s con juga te to J N and c i s of type k N I . Proceed as in 1.1) . Case 1.5) X |A=ax id . Then X i s con juga te to J 2 C or j Q . j 2 c i s not p o s s i b l e s i n c e [S] = X*[S] = J 2 C * ( [ Z j + a [ M ] ) = -[L]-a[M] = - [S ] , So X i s con jugate to j Q and t i s of type k Q F . Le t B,=B and B 2 = 9V - ( A u d ( A ) U B ) . Let J be a nonsepa r a t i ng 1-ce l l of A wi th J n XJ= 0 and l e t 1^  be any path in B^  from 9J to d ( 9 J ) . B^/t i s an annulus so by l i f t i n g an embedded path tha t i s path homotopic to I^ /i we may a l s o assume that l^ n c l^ = 0 . By making proper c h o i c e s , we ar range tha t I , u I 2 u J u d ( J ) bounds a 2 - c e l l i n V. A s i m i l a r p r o p e r t y ho lds f o r X' f o r J ' = h ( J ) . Use I '^ and 1^' to extend h to a c o n j u g a t i o n 9V >9V and complete the argument as b e f o r e . Case 2) iA n A= 0 and tW,.=W+. Let S be a f i x e d component of 9A. Le t B be a component of W + n 9 W tha t meets S. Let X=t|W + . There are two p o s s i b i l i t i e s : 2a) XB=B 70 2b) XB NB= 0 . In 2a) i i n t e r changes boundary components of 3W, wh i le in 2b) i t does no t . W+ i s a s o l i d t o r u s , so X i s con juga te to one of the s tandard i n v o l u t i o n of the s o l i d t o r u s . T h i s s p l i t s the p resen t case i n t o four subcases . In f a c t i n case 2a) i f i i s o r i e n t a t i o n p r e s e r v i n g then X i s con juga te to J 2 c * i f i i s o r i e n t a t i o n r e v e r s i n g then X i s con juga te to j or J N » i n case 2b) i f i i s o r i e n t a t i o n p r e s e r v i n g then X i s con juga te to j g or j Q . i f c i s o r i e n t a t i o n r e v e r s i n g then X i s con juga te to J 2 p . To show t h i s note tha t in H,(W + )=Z , [S] i s a g e n e r a t o r . S ince S £ 3W + -Fix, X cannot be con jugate to J 2 D r J M or J D p« In case 2a) X * [ S ] = ~ M ( X ) [ S ] , wh i l e in case 2b) X * [ S ] = M ( X ) [ S ] , where M (X ) i s +1 i f X i s o r i e n t a t i o n p r e s e r v i n g and -1 i f X i s o r i e n t a t i o n r e v e r s i n g . I f X i s con juga te to J 2 C or j 2 p then X [ S ] = - [ S ] . In a l l o ther cases X [ S ] = [ S ] . T h i s e s t a b l i s h e s the c l a i m . X .=i|W. must s a t i s f y the ( s i m i l a r ) case 2a) - 2b ) . Combining X and X. in a l l the d i f f e r e n t p o s s i b l e ways g i v e s i n v o l u t i o n s of types as l i s t e d p r e v i o u s l y . For example, combin ing a J A wi th a JN< g i v e s an i n v o l u t i o n of type k^. 71 Let i ' be of same type as i . I t remains to show they are c o n j u g a t e . F i n d an isomorphism h:A >A' and extend by e q u i v a r i a n c e to h : A u i A >A' y i ' A ' . It s u f f i c e s to show that h extends to W+ when X=i|W+ and X' = t ' |W + are c o n j u g a t e . Take B and B' as above. Case 2.1) X i s con juga te to J A < F i x n B i s a 1-sphere and the components of B-Fix are i n t e r c h a n g e d . Extend h over one of these components and then extend over a l l of B by e q u i v a r i a n c e . S i m i l a r l y f o r the o ther annulus of W + n 9 W . By the c o n j u g a t i o n ex tendab le p r o p e r t y of J a t h i s c o n j u g a t i o n extends to a l l of W +. Case 2.2) X i s con juga te to J N . Then B/t i s a Mobius band. The isomorphism h extends to B/t. L i f t to B and proceed as in 2 . 1 . Case 2.3) X i s con juga te to J 2 C - B / / t ^ s a 2 - c e l l w i th F i x n B / t be ing two p o i n t s . There i s an isomorphism which extends the g i ven induced one on 9B/i and takes the two p o i n t s of F i x n B / i to F i x ' n B ' / i ' in e i t h e r of the two p o s s i b l e ways. T h i s isomorphism l i f t s and w i th c o r r e c t c h o i c e s h extends as in 2 . 1 . Case 2.4) X i s con jugate to J 2 p - Extend h in any way to B and then extend by e q u i v a r i a n c e to 9W+. Case 2.5) X i s con juga te to j Q or j g . Then B n XB=0. Let J be a proper 1-ce l l of A and l e t J ' = h ( J ) . S e l e c t a proper 1-ce l l I of B wi th 9I = 9 ( J u X J ) n A and c o n s i d e r 72 C=I u XI |_j J u XJ. C cannot be used to extend h s i n ce even i f C bounds a d i s c in W +, C may not bound a d i s c in W' + . As be fo re l e t W+ = D 2 x S 1 , M=S1x1 and L=1xS 1 . Then [M] and [L] genera te H, (9W + )=Z©Z and wi th c o r r e c t c h o i c e s [S]=[L]+a[M] as c l a s s e s in H 1 (3W + ) , f o r some aeZ. By changing I assume [C] = A i [ L ] + b[M] where u i s 0 or 1 and beZ. Ach ieve t h i s by a l t e r i n g the path c l a s s of I by c o n c a t e n a t i n g w i th S and by us i ng the f a c t that X* [S ]=[S ] , Now W+/X i s a t o r u s . Let p:H,(9W + ) >H!(9W+/X) be the obv ious homomorphism. Let [A*,] and [Ly] be gene ra to r s fo r H 1 (9W +/X) d e f i n e d as f o r 9W+. Without l o s s f o r j Q , p[M]=[M,] and p [L ]=2 [L , ] and fo r j g , p[M]=2[M,] and p [ L ] = [ L , ] . For j s , p [ C]=M [ L ,]+2b[M y ]. S ince C/X i s double covered by C i t f o l l o w s u i s even . So y=0. Let C,=C. For j Q , suppose M i s 0. Then p[C]=b[Atf,] so b i s even s i n c e aga in C/X i s double covered by C . Then [ (I u J )/X] = (b/2) [My ] and i t f o l l ows ( I U J ) / X l i f t s to a 1-sphere. T h i s i s a c o n t r a d i c t i o n s i n ce 9 ( I U J ) # 0 . So AI=1. Then l e t I, be a proper 1-ce l l in B wi th I , n 1 = 91, = 91 such tha t C , = 1 , u XI u J |_j XJ has c l a s s [C 1]=d[j^] f o r some deZ. In any event we o b t a i n a curve C, w i th [C , ]=0eH, (W + ) . S ince jg and determine d i f f e r e n t u i t f o l l o w s that f o r i' we can d e f i n e C , ' in the same way as C, ( i . e . ) us ing I,' i f 73 C, uses I,. Extend h over I (and I, f o r c a s e ) . T h i s h then extends to a c o n j u g a t i o n by c o n s t r u c t i o n . Case 3 ) iA n A= 0 and tW + =W. . Then c i s f i x e d p o i n t f r e e . Suppose i ' i s of same t ype . Let h:A >A' be any isomorphism and extend by e q u i v a r i a n c e to a c o n j u g a t i o n h r A L j i A > A ' | _ j i ' A ' . F i x a component S of 9A and l e t S '=h (S ) . Let B be the component of W+ n3W that meets S. B i s an annu lu s . S i m i l a r l y d e f i n e B'. S ince i and i ' have the same in t e r change type (F/I p rope r t y ) we have h (9B-S)=9B '-S ' . S ince they have the same o r i e n t a t i o n type h|3B extends to h:B >B ' . The isomorphism determined on the annulus A u B u i A n e c e s s a r i l y extends to an isomorphism of the s o l i d t o r i W + >W + ' . Extend to W=W + UW. by e q u i v a r i a n c e . QED C o r o l l a r y 7.4 Let W and W be t r i v i a l I-bundles over a t o r u s . I n v o l u t i o n s con jugate to k T are c o n j u g a t i o n e x t e n d a b l e . If t on W i s con juga te to k A or k 2 A and c' on W i s con juga te to i then a c o n j u g a t i o n h:3W >3W i s c o n j u g a t i o n ex tendab le i f i t s a t i s f i e s the f o l l o w i n g c o n d i t i o n : Let F i x , be a component of F i x = F i x ( i ) and l e t F i x , x [ - 1 , 1 ] be a b i c o l l a r of F i x , = F i x x O such that 9 F i x , x [ - 1 , l ] b i c o l l a r s 3 F i x , . S i m i l a r l y f o r F i x , ' , where F i x , ' i s a component of F i x ' = F i x ( t ' ) meet ing h ( 9 F i x , ) . Then r e q u i r e that h extends to an 74 i somorphi sm h : 9 W u F i x , x [ - 1 , 1 ] >9W u F i x , ' x [ - 1 , l ] . There are c o n j u g a t i o n s of 9W that a re not ex t endab l e ! P r o o f : For k T , the f i x e d set s e p a r a t e s . Let i on W be con juga te to k T < Let W + be the c l o s u r e of one of the components. Then W+ i s i somorph ic to T X [ 0 , 1 ] by an isomorphism t a k i n g 9W +-Fix to TxO and F i x to Tx1 where T = S 1 x S 1 . C l e a r l y the isomorphism h:TxO >T'xO extends to an isomorphism on W+ t a k i n g Tx1 to T ' x 1 . Extend by equ i va r i a n c e . For k A , l e t i, i ' and h be as in statement of c o r o l l a r y . I t f o l l o w s h extends to a c o n j u g a t i o n h : 3 W u F i x > 9 W ' u F i x ' . C u t t i n g W open a long F i x g i v e s a s o l i d t o rus V hav ing two c o p i e s of F i x in i t s boundary . The i n v o l u t i o n t on W i s induced by an i n v o l u t i o n X on V which i n t e r changes these c o p i e s of F i x . S i m i l a r l y f o r W . By the c o n d i t i o n on the b i c o l l a r , . h|(9W|_jFix) i s induced from a c o n j u g a t i o n h^av >9V . Now X i s con juga te to J N so by the c o n j u g a t i o n ex tendab le p rope r t y fo r J n, h, extends over V and hence induces a c o n j u g a t i o n on W ex tend ing h. For k 2 A ' * e t  L' l '  an^  1 1 k e a S * n s t a t e m e n t o f c o r o l l a r y . I t f o l l o w s h extends to a c o n j u g a t i o n h : 9 W u F i x , > 9 W u F i x , ' . S ince a l l components of F i x and 3 W - F i x are a n n u l i h extends to a c o n j u g a t i o n h : 3 W u F i x > 3 W u F i x ' . Le t C and tC be the two 3-ce l l s that F i x decomposes W i n t o . By the b i c o l l a r c o n d i t i o n h(3C) i s c o n t a i n e d in one of the two 3 - c e l l s that F i x ' decomposes W i n t o . Say h(3C) c 3 C . Then extend h to an isomorphism h : W u C > W ' L J C ' by con ing to a ve r tex and extend by e q u i v a r i a n c e to the d e s i r e d c o n j u g a t i o n . QED C o r o l l a r y 7 . 5 If i i s an o r i e n t a t i o n p r e s e r v i n g i n v o l u t i o n on W=S 1xS 1xI then W/1 i s i somorph ic to one of the f o l l o w i n g spaces : W/k 2 s = D 2 x S 1 a s o l i d t o rus wi th F i x / k 2 S two unknot ted 1-spheres ( ± 1 / 2 ) X S 1 , W/kg = D 2 x S 1 a s o l i d t o rus wi th F i x / k g one unknot ted 1-sphere { ( e 7 r i t / 2 , e 2 , r l t ) :-1<t<1 } r e p r e s e n t i n g twice a genera to r of H , ( D 2 x S 1 ) , W/k^j an o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e , W/k 4 C = S 2 x l wi th F i x / k 4 C = {four p o i n t s } x I . w/kQF = W. Proof: i i s con jugate to a s tandard i n v o l u t i o n k. Use the r e p r e s e n t a t i o n s fo r the s tandard i n v o l u t i o n s . In a l l cases except f o r k o p , W/k = S 1 x { x + y • i : 0 < y } x l / ( g u g ' ) where g i s an i d e n t i f i c a t i o n of S ' x l x l and S 1 x - 1 x l depending on k. S 1 x { x + y i : 0 < y } x O / ( g u g ' ) i s a F i x / k c . g ' i s an i d e n t i f i c a t i o n of For kg note tha t Mobius band w i th boundary QED #8. I n v o l u t i o n s on the O r i e n t a b l e I-Bundle Over a K l e i n  B o t t l e D e f i n i t i o n 8.1 Let W=S 1xIxI/d be the o r i e n t a b l e tw i s t ed I-bundle over the K l e i n b o t t l e S 1 x l x 0 / d , where d=/cx( r | 31 ) X T . More e x p l i c i t l y , W=S 1 x[-1,1]x[- 1 ,1 ]/ (z ,- 1 , t ) ~ ( z , 1 , - t ) . z = ±i i s a s e p a r a t i n g annu lus , whereas z=T i s a nonsepa ra t i ng Mobius band. See F i g u r e 11. The I - f i b e r s are z x s x l . An i n v o l u t i o n X on S ' x l x l w i th X | .d=d.X | where X| denotes X ^ S ^ B I x l ) induces an i n v o l u t i o n k=X/d on W. F i x = F i x ( k ) = ( F i x ( X ) u F i x ( d _ 1 . X | ) ) / d . D e f i n e the f o l l o w i n g i n v o l u t i o n s on W (see F i g u r e 11) : k = i d x i d x r / d hav ing f i x e d set a K l e i n b o t t l e S 'x IxO/d k „ =icxidxid/d hav ing f i x e d set two Mobius bands ± 1 x l x l / d 7 7 k 2 g=Kxidxr / d hav ing f i x e d set two 1-spheres ±1xlx0/d k A =(-/c)x idxid/d hav ing f i x e d set an annulus ± i x l x l / d k g =(-K ) x i d x r / d hav ing f i x e d set a 1-sphere ± i x l x 0 / d k A 2 p = i d x T x i d / d hav ing f i x e d set an annu lus and two p o i n t s (S 'xOxI u ± 1 x - 1 x 0 ) / d k g 2 Q = i d x r x r / d hav ing f i x e d set a 1-sphere and two 1-ce l l s ( S 1 x O x O u ± 1 x - 1 x l ) / d k 2 ^=(-«)xTxid/d hav ing f i x e d set two 1-ce l l s ± i x 0 x l / d ^2p=(~K)xTxr/d hav ing f i x e d set two p o i n t s ±ixOxO/d k^=axidxid/d f i x e d p o i n t f r ee and o r i e n t a t i o n p r e s e r v i n g k N = a x i d x r / d f i x e d p o i n t f r e e and o r i e n t a t i o n r e v e r s i n g These e l even i n v o l u t i o n s are not con jugate because t h e i r f i x e d p o i n t se t s or o r i e n t a b i l i t y types are d i f f e r e n t . R e c a l l tha t the i n v o l u t i o n s wi th even d imens iona l f i x e d p o i n t s e t s are o r i e n t a t i o n r e v e r s i n g and tha t the ones w i th odd d imens iona l f i x e d se t s are o r i e n t a t i o n p r e s e r v i n g s i n c e W i s o r i e n t a b l e . Lemma 8.2 Let W=S 1 xIxl/d be the o r i e n t a b l e I-bundle over a K l e i n b o t t l e . Then i f A i s an e s s e n t i a l annulus the re i s an ambient i so topy moving A so tha t A i s of form S ' x - l x l / d i f A i s nonsepa ra t i ng or of form ± i x l x l / d i f A i s s e p a r a t i n g . F i g u r e 1 1 . Fixed point sets for the standard involutions. S 2 x - l x l 79 P roo f : Remove components of A n , ( S 1 x i x O ) / d . For d e t a i l s see [11 ] . QED Theorem 8.3 Let i arid t' be i n v o l u t i o n s on the o r i e n t a b l e I-bundle over a K l e i n b o t t l e , W=S 1 xIxl/d where d = « x ( T | 9 1 ) X T . Suppose i and t' have i somorphic f i x e d se t s and i f t and t' are f i x e d p o i n t f r ee assume, in a d d i t i o n , tha t they have the same o r i e n t a t i o n t ype . Then i and t' are c o n j u g a t e . An i n v o l u t i o n on W i s con juga te to one of the e leven i n v o l u t i o n s l i s t e d above. P r o o f : The proof i s s i m i l a r to the proof of Theorem 7 .3 . Let t be an i n v o l u t i o n on W=S 1 xIxI/d. We show i t i s con juga te to a s tandard i n v o l u t i o n . By the Annulus Theorem 5.9 the re i s an e s s e n t i a l annulus A wi th e i t h e r iA n A= 0 or iA=A and A and F i x t r a n s v e r s a l . In the l a t t e r case by Lemma 3 .5 , assume the c o l l a r of A i s not i n t e r c h a n g e d . By the p r e v i o u s lemma take A to be nonsepa ra t i ng of form S 1 x - 1 x l / d or s e p a r a t i n g of form ± i x l x l / d . In the case where A i s nonsepa ra t i ng and iA n A= 0 make iA=S 1 xOxI . Let W+=S 1 x [ 0, 1 ]xl and W . = S ' x [ - 1 , 0 ] x l . 80 There a re f i v e c a s e s : Case 1) iA=A, A i s n o n s e p a r a t i n g , A and F i x are t r a n s v e r s a l and the c o l l a r of A i s not i n t e r c h a n g e d . Case 2) iA nA = 0 , A i s nonsepa ra t i ng and iW t=W +. Case 3) iAr-|A=0, A i s nonsepa ra t i ng and iW+=W.. Case 4) tA=A, A i s s e p a r a t i n g , A and F i x are t r a n s v e r s a l and the c o l l a r of A i s not i n t e r c h a n g e d . Case 5) iA[-jA=0 and A i s s e p a r a t i n g . We show that t i s con juga te t o : in case 1) k R , k 2 M , k 2 g , k A , k g , k Q or k N in case 2) k A 2 p , k g 2 c , k 2 c or k 2 p i n case 4) k R , k 2 M , k 2 g , k A 2 p or k s 2 c i n case 5) k A , k g , k 2 c , k 2 p , k Q or k N and that case 3) does not a r i s e . S e ve ra l of the s tandard i n v o l u t i o n s a re l i s t e d in more than one c a s e . Each s tandard i n v o l u t i o n (or at l e a s t a con juga te of one) can in f a c t a r i s e in the case i t has been l i s t e d under . To see t h i s i t s u f f i c e s to d i s p l a y an annulus A' in W wi th p r o p e r t i e s ana logous to those of A. Cons ide r A' as f o l l o w s : in. case 1) take S ' x - l x l / d ; in case 2) take S 1 x ( - 1 / 2 ) x l / d ; in case 4) take ± i x l x l / d ; in case 5) take e ± i 7 r / 4 x l x l / d . I t s u f f i c e s to show, f i r s t , tha t t has the same ( f i x e d se t ) type as a s tandard i n v o l u t i o n l i s t e d under a 81 c o r r e s p o n d i n g c a se , and second , i f i and t' have the same type and f a l l i n t o the same case 1) - 5) then they are c o n j u g a t e . Constant use i s made of S e c t i on 4. Reserve j to denote s tandard i n v o l u t i o n s on the s o l i d t o r u s . A l l c o n s t r u c t i o n s done fo r i are to be per formed fo r i ' , even i f not e x p l i c i t l y s t a t e d . Case 1) Proceed as fo r Case 1) of Theorem 7 .3 . The i d e n t i f i c a t i o n i s now d = « x ( r | ) x T i n s t ead of d = i d x ( T | ) x i d . Thus two of the f i v e p o s s i b i l i t i e s fo r X|A g i ve d i f f e r e n t f i x e d s e t s . When X|A=/<xid we have F ix (X ) i s two 2 - c e l l s . Then F i x i s e i t h e r two Mobius bands or one a n n u l u s . When X|A=idxr we have F i x (X ) i s an annu lus . Then F i x i s a K l e i n b o t t l e . Case 2) i A n A = 0 , A nonsepa ra t i ng and iW+=W+. S e l e c t the component S=S 1x-1x-1 of 9A. Let B+ be the component of W + n 9 W tha t meets S. Let X + =i |W + . There are two poss i b i 1 i t i e s : 2a) X + B + =B t 2b) X + B + n B + =0. S i m i l a r l y f o r B. and X .=t|W. . Suppose X+ s a t i s f i e s case 2a ) . Then X + ( S )=S 1 xOx-1. S ince (X. |A ) .d=X + |A e v a l u a t i n g at S g i v e s X . ( S ' x1x1 )=S 1 xOx-1 . T h e r e f o r e X. s a t i s f i e s case 2b ) . S i m i l a r l y i f X+ s a t i s f i e s case 2b) then X. s a t i s f i e s case 2a ) . The conjugacy c l a s s of X i s a l s o r e s t r i c t e d by the 82 o r i e n t a t i o n type of i . Up to symmetry the re are four c a s e s : Case 2. 1 ) x + i s con juga te to j 2 c and X. i s con jugate to Case 2. 2) *• i s con jugate to j 2 c and X. i s con jugate to Case 2. 3) x + i s con juga te to J A and X. i s con juga te to J 2 P -Case 2. 4) x* i s con jugate to J N and X. i s con jugate to J 2 P -These g i ve r i s e to i n v o l u t i o n s wi th f i x e d se t s as c l a imed f o r t h i s c a s e . The case i s completed as case 2) in Theorem 7.3. Case 3) iA n A = 0 , A nonsepa ra t i ng and iW + =W. . Take S=S 1xIxO w i th some c h o i c e of o r i e n t a t i o n . Then S i s a genera to r of H,(W+) and H,(W.) . Now t | ( W . n S ) = d . i | ( W + n S ) but d* [S ]=-[S ] . T h i s i s a c o n t r a d i c t i o n so t h i s case cannot a r i s e . Case 5) iA n A= 0 and A s e p a r a t e s . I t f o l l o w s tha t IA a l s o sepa ra tes and that cA i s c o n t a i n e d in one of the two components that A decomposes W i n t o . By a s u i t a b l e i somorph ism, we may assume A = e ± ^ 7 f / / ' 4 x l x l / d and c A = e ± ^ 7 r / / 4 x I x l / d . A and tA decompose W i n t o th ree s o l i d t o r i components U 0 , U , , U 2 w i th U i n U 2 = 0 , U 0 n u i = A ' U 0 n u 2 = t A and tU,=U 2 and tU 0 =U 0 . Moreover , i f S i s a component of 3A then 83 [ S ] eH 1 (U 0 ) i s a genera to r and [S leH^dJ , ) i s twice a g e n e r a t o r . X= i |U 0 i s an i n v o l u t i o n on a s o l i d t o rus that i n t e r changes the d i s j o i n t a n n u l i A and iA and both a n n u l i have boundar i es r e p r e s e n t i n g a genera to r of H , ( U 0 ) . T h i s i s the same s i t u a t i o n as f o r X in case 2) of Theorem 7 .3 . That argument showed X i s con jugate to J 2 c ^A' -*N' ^S' -'O O R j 2 p . S ince i has the same f i x e d set as i | U 0 we o b t a i n f i x e d se t s as l i s t e d above. Suppose i ' a l s o f a l l s i n t o t h i s c a s e . Then s e l e c t an isomorphism h:A >A' which we extend by e q u i v a r i a n c e to a c o n j u g a t i o n h : A u i A >A' u iA ' . The arguments fo r case 2) in Theorem 7.3 show h extends to a c o n j u g a t i o n h : U 0 > U 0 ' . The f o l l o w i n g c l a i m shows h|A extends to an isomorphism h:U, > U , ' . Extend to U 2 by e q u i v a r i a n c e o b t a i n i n g a c o n j u g a t i o n h:W >W between t and i ' and c o n c l u d i n g t h i s c a s e . C l a i m : Let U be a s o l i d t o rus and A an annulus in 3U. Suppose a component S of 9A r ep r e sen t s tw ice the genera to r of H , (U ) . S i m i l a r l y f o r A ' in U ' . Then an isomorphism h:A >A' extends to an isomorphism h:U >U ' . To prove t h i s , choose 1-spheres M and L in 3U so tha t [M] and [L] generate H,(3U) and [M] i s t r i v i a l in 14,(1!). The cho i c e can be made so tha t [S ]=a[M]+2 [L ] where " a " i s an odd i n t e g e r . Let I be a proper 1-ce l l in A meet ing both boundary components of A. S i m i l a r l y f o r U ' , u s i ng I '=h ( I ) . B=3V-A i s an a n n u l u s . There i s a proper 1-ce l l J of B w i th 3J=3I. Le t 84 S ^ I L J J . Then [ S , ] =b[A)]+c [ L ] where " c " i s an odd i n t ege r s i n c e Sp i S , i s a p o i n t . There i s an isomorphism of 3U to i t s e l f l e a v i n g A f i x e d which changes [S , ] by a g i ven m u l t i p l e of [ S ] , So we may choose J so tha t [S,]=d[M]+[L] f o r some deZ . Choose J ' s i m i l a r l y . Extend h|A to an isomorphism h : A u J > A U J ' . S ince in t (B- J ) i s an open 2 - c e l l , h can be extended by con ing to an isomorphism h : 3 U > 9 U ' . Then (2d-a)h*[Al] = h*(2 [S,]-[S ]) = 2 [ S , ' ] - [ S ' ] (2d'-a ' ) [> ' ] . S ince a and a ' are odd and [M] and [M'] are gene ra to r s we get hil[M] = ±[M']. Hence h extends to h:U >U' . Case 4) iA=A, A i s s e p a r a t i n g , A and F i x are t r a n s v e r s a l and the c o l l a r of A i s not i n t e r c h a n g e d . Let W + ={z:z=x+y•i ,x>0}x lx l/d and W . = { z : z = x + y « i , x < 0 } x l x l / d . Then iW+=W+ and W + and W. are s o l i d t o r i . Let S be a component of 3A. X + =i|W + i s con juga te to a s t anda rd i n v o l u t i o n j of a to rus D 2 x S 1 . Let M= S 1x1 and L=1xS 1 . Then on choos ing c o r r e c t o r i e n t a t i o n s [S]=a[Af] + 2[L] e H , ( 3 ( D 2 x S 1 ) ) where a i s odd . S ince iA=A i t f o l l o w s X + *[S ] = / u[S] where M=±1 and u depends on ly on i |A . Check ing these c o n d i t i o n s f o r the s tanda rd i n v o l u t i o n s on a s o l i d t o rus g i v e s : u=1 and X + i s con jugate to j ^ , J s or J M . M=-1 and X+ i s con jugate to J2 C or J D P . S i m i l a r l y f o r X.=t|W. . The c o l l a r i s not i n t e r changed so i 85 and t|A have the same o r i e n t a b i l i t y t ype . We o b t a i n four c a s e s : Case 4.1) X + and X. are con juga te to J M and t|A i s con juga te to i d x r or a x r . Case 4.2) X+ and X. are con juga te to j ^ p and t |A i s con juga te to >cxid. Case 4.3) X+ and X. are con juga te to jg or j ^ and c|A i s con juga te to a x i d . Case 4.4) X+ and X. are con juga te to j 2 c and i. [ A i s con jugate to K X T . For case 4.1) we w i l l show the isomorphism c l a s s of the f i x e d set determines t |A . We w i l l a l s o show the d i f f e r e n t cases 4.1) - 4.4) have non isomorph ic f i x e d s e t s . T h e r e f o r e g i ven t' w i th f i x e d set i somorph ic to tha t of t, t he re i s a c o n j u g a t i o n h:A >A ' . We show h extends to a c o n j u g a t i o n h:W+ >W + 1 . S i m i l a r l y h extends over W. and the p roof w i l l be comp le te . Let B= 3W,. -A. B i s an annu lu s . In case 4.1) t|A i s con juga te to i d x r or axr. In the f i r s t case F i x n A = S 1 so F i x i s a K l e i n b o t t l e wh i le in the second case F i x n A = 0 so F i x i s two Mobius bands. In the f i r s t case B/t i s a Mobius band. I t f o l l o w s h:A >A' extends to a c o n j u g a t i o n h:9W,. >3W + ' . In the second case B/t i s an annulus w i th 9 ( B / t ) = ( 9 A / t ) u ( 9 F i x n W + ) / t . Extend h/t and l i f t to a c o n j u g a t i o n h:9W + >3W + ' . In both cases the c o n j u g a t i o n ex tendab le p r o p e r t y of j M shows t h i s 86 c o n j u g a t i o n extends to h:W+ >W + ' . In case 4.2) l e t F i x + denote the f i x e d 2-disc component of A* . Let F i x + x [ - 1 , l ] be a b i c o l l a r of F i x + . S ince a component of 3A has i n t e r s e c t i o n number ±2 wi th F i x + i t f o l l o w s F i x + x1 meets both components of A - F i x . S i m i l a r l y f o r F i x . x l . Hence F i x + u F i x . i s b i c o l l a r e d so i t must be an a n n u l u s . Thus the f i x e d set of i i s an annulus and two p o i n t s . Extend the c o n j u g a t i o n to A u 9 F i x + >A' u 3 F i x + ' . S ince 3W + - ( 3 F i x + u A ) i s two open 2 - c e l l s tha t are i n t e r changed under t we can extend to a c o n j u g a t i o n 3W+ >3W+' and so by the c o n j u g a t i o n ex tendab le p rope r t y of J D P to W +. In case 4.3) l e t S be a f i x e d component of A. W + / i i s a s o l i d t o r u s . Let p:H,(W + ) >H,(W + /i) be the obv ious homomorphism. [S]=a[M]+ 2[L] where a i s an odd i n t e g e r . F i r s t we show j Q i s not p o s s i b l e . Compare w i th case 2.5) of Theorem 7 .3 . p[S]=a[M y ] + 4 [L , ] . However, S double cove r s S/t so i t f o l l o w s a i s even, a c o n t r a d i c t i o n . .So on ly j s o ccu rs and F i x i s two 1-spheres. Then p[S] = 2a[A4, ] + 2 [L , ] and [ S/i ] =a [ M, ] + [ L, ] . Let I be a proper 1-ce l l in A that meets both boundary components of A such tha t I n t l=0 . S i m i l a r l y f o r I ' = h ( l ) . There i s a proper 1-ce l l J , in W+/t w i th 3 J ,=3( I/ i ) and [ J , u I ] = ± [ M , ] . Let S 0 = J U t J u I |j t l where J i s a l i f t of J , by p 1 . Then [S0]=±[M]. S i m i l a r l y f o r t ' . Extend h:A >A' to h : A|_jJ >A u J' and then by e q u i v a r i a n c e to h : A u J u t J >A' u J ' |_j t ' J ' . Now 3W + -(A u J u c J ) c o n s i s t s of two 2 - c e l l s tha t are i n t e r changed under t . So h extends to h:3W >3W . The c o n d i t i o n on [S 0 ] and the c o n j u g a t i o n ex tendab le p rope r t y of j g show h extends to a c o n j u g a t i o n on W. In case 4.4) X + has f i x e d set F i x , + u F i x 2 + where F i x 1 + and F i x 2 + are proper 1-ce l l s of W +. Arguments s i m i l a r to those g i ven a l r eady show that F i x ^ + n A cannot be e x a c t l y one p o i n t , i = 1 , 2 . Say then tha t 3 F i x 2 . c A . S i m i l a r l y f o r X . . Then the f i x e d set of t i s two 1-ce l l s F i x 1 + , F i x , , and one 1- sphere F i x 2 + u F i x 2 . . • (3W + -A)/t i s a 2 - c e l l and h/t i s g i ven on the boundary . C l e a r l y h/t can be extended over the 2- c e l l . On l i f t i n g o b t a i n a c o n j u g a t i o n h:3W + >3W + ' . S ince F i x n A are two p o i n t s in the same component of Fixr-|W + and-s i n ce a l s o h ( F i x n A ) = F i x ' n A ' are in the same component of F i x ' n w \ ' the c o n j u g a t i o n ex tendab le p r o p e r t y of J 2 C g i v e s a c o n j u g a t i o n h:W+ >W + ' . QED C o r o l l a r y 8.4 On the o r i e n t a b l e I-bundle over a K l e i n b o t t l e i n v o l u t i o n s wi th 2-d imens iona l f i x e d se t s are c o n j u g a t i o n e x t e n d a b l e . P r o o f : For k A 2 p the c o n j u g a t i o n extends over the f i x e d s e t . Then cut open on the f i x e d set and use the c o n j u g a t i o n ex tendab le p r o p e r t y of the s o l i d t o rus i n v o l u t i o n J2p* T n e other cases are s i m i l a r to those f o r the t r i v i a l I-bundle over a t o r u s . QED C o r o l l a r y 8.5 If t i s an o r i e n t a t i o n p r e s e r v i n g i n v o l u t i o n on an o r i e n t a b l e tw i s t ed I-bundle W=S 1 x Ix l /d over a K l e i n b o t t l e then W/t i s i somorph ic to one of the f o l l o w i n g spaces : W/k 2 S = D 2 x S 1 a s o l i d t o rus w i th F i x / k 2 S two unknotted 1-spheres ± 1 / 2 X S 1 , W/kg = D 2 x S 1 a s o l i d t o rus w i th F i x / k g one unknot ted 1-sphere { (e n l t/2,e 2* 1 1): - 1<t<1} r e p r e s e n t i n g twice a genera to r of H , ( D 2 x S 1 ) , W / k s 2 C = D 2 x l a 3 - c e l l w i th F i x / k s 2 c = ±1/2x1 u ( ( 3 / 4 ) ( z / | z | ) ) x 0 two 1-ce l l s and one l i n k e d 1-sphere, W/k 2 c = D 2 x l / ( a | 3 D 2 ) X T an o r i e n t a b l e I-bundle over a p r o j e c t i v e p lane wi th Fix/k 2 ^,= ( ± l / 2 ) x l two 1-ce l l f i b e r s , W/kQ = W. P r o o f : i i s con juga te to a s t andard i n v o l u t i o n k. Use the r e p r e s e n t a t i o n s fo r the s tandard i n v o l u t i o n s . W/k a r i s e s from the f o l l o w i n g subspaces of W by i d e n t i f i c a t i o n s on 89 t h e i r b o u n d a r i e s : ( x + y i : 0 < y } x l x l f o r k 2 g and k Q , S ' x l x [ 0 F 1 ] f o r kg , and S 1 X [ 0 , 1 ] X I f o r k s 2 c and k 2 C . QED 90 V. INVOLUTIONS ON ORIENTABLE TORUS BUNDLES OVER A 1-SPHERE  AND ON UNIONS OF ORIENTABLE TWISTED I-BUNDLES OVER KLEIN  BOTTLES £9. I n v o l u t i o n s With 1-Dimensional F i x e d Sets Let g : T 2 >T 2 be an isomorphism where T 2 = S 1 x S 1 and l e t d :T 2 x-1 >T 2x1 be d e f i n e d by d ( x , - 1) = ( g ( x ) , 1 ) . De f i ne the t o rus bundle M by M = T 2 x I / d . Then M i s i r r e d u c i b l e and g g g T 2 x1 i s a nonsepa ra t ing i n c o m p r e s s i b l e 2-s ided t o r u s . Up to i so topy g : T 2 >T 2 ' i s un i que l y determined by g * : H , ( T 2 ) > H , ( T 2 ) . Let S ^ S ' x l and 52=1x5 ' . Then wi th r e spec t to the b a s i s [ S , ] , [ S 2 ] of H ^ T 2 ) , g* i s g i ven by a mat r ix M(g) of G L 2 ( Z ) . The mat r ix w i th r e spec t to a d i f f e r e n t b a s i s of H , ( T 2 ) i s a con juga te Q 1 M(g) Q of M(g) , where Q e G L 2 ( Z ) . M i s o r i e n t a b l e i f and on l y i f g i s y o r i e n t a t i o n p r e s e r v i n g , g i s o r i e n t a t i o n p r e s e r v i n g when and on l y when the det (M(g ) )=1. 91 Of i n t e r e s t are the o r i e n t a b l e f l a t space forms M 1 r ' « ' , M 5 . See [15 ] . These are determined by g as f o l l o w s : M ^ S ' x S ' x S 1 : g= id . Then M(g)= J ?] • . M 2 : g = K X K ( i . e . ) g (x ,y) = (x ,y) . Then M(g)= |^  ^ . M 3 : g=cj. ( K x i d ) . p ( i . e . ) g (x , y) = (y, xy) . Then M(g)= M „ : g = c j c ( ) c x i d ) ( i . e . ) g (x , y) = (y, x) . Then M(g)= Q j . M 5 : g=w.p .(Kxid) ( i . e . ) g ( x , y ) = ( y , x y ) . Then M(g)= ^ } j . Each of these spaces has i n v o l u t i o n s w i th 1-dimensional f i x e d s e t s . Let W, and W2 be two o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . An isomorphism d:9W, >9W2 determines a un ion a long the boundar i e s of two o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e M^=(W, u W 2 ) /d . M^ i s double cove red by an o r i e n t a b l e t o rus bundle over S 1 . i s i r r e d u c i b l e . Let T 2 =3W,. The nonsepa ra t i ng a n n u l i of W, determine a c a n o n i c a l genera to r (1,0) of H ! ( T 2)=Z©Z up to s i g n . S epa ra t i ng ( n o n t r i v i a l ) a n n u l i or Mobius bands of W, determine a genera to r (0,1) up to s i g n . Note that the i n v o l u t i o n s k 2 M and k A 2 p o n W l a r e i s o m o r P b i s m s of W, tha t r e ve r se the s i gns of these g e n e r a t o r s . As b e f o r e , the isomorphism d determines a mat r ix of G L 2 ( Z ) . 92 An a l t e r n a t e d e s c r i p t i o n f o r these spaces i s M = T 2 x l / ( d . ( x ) , -1 )~ (x ,-1 ) , ( d + ( x ) , 1 ) ~ ( x , 1 ) where d . and d + are f i x e d po in t f r e e o r i e n t a t i o n r e v e r s i n g i somorph isms . Then T 2 xO decomposes M i n t o two o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Note tha t M 2 i s a union of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e : S 1 x±1x l are two K l e i n b o t t l e s . The o r i e n t a b l e f l a t 3-dimens iona l space forms have been c l a s s i f i e d (see Wolf [ 15 ] ) . Up to a f f i n e equ i v a l ence the re are on l y s i x such space forms M , , ' - * , ! ^ and M 6 . De f i ne M 6 by M 6 = S ' x S ' x I / ( x , y , - 1 ) ~ ( - x , - y , - 1 ) , ( x , y , 1 ) ~ ( - x , - y , 1 ) M 6 i s a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e but i s not a t o rus bundle s i n ce H 1 ( M 6 ) = Z 2 © Z 2 i s f i n i t e . M 6 i s a l s o known as the Hantzsche-Wendt m a n i f o l d (see [ 4 ] ) . We need two lemmas which d e s c r i b e the p o s i t i o n of i n c o m p r e s s i b l e t o r i in M. For d e t a i l s see [ 11 ] , Lemma 9.1 Let M be an o r i e n t a b l e t o rus bundle over S 1 and l e t T be an i n c o m p r e s s i b l e t o rus in M. If T i s nonsepa ra t i ng then M = T 2 x [ - 1 , l ] / d where d :T 2 x-1 >T 2x1 i s an i somorphism. If T i s s e p a r a t i n g then T decomposes M i n t o W, and W 2 , two o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . M=W, U W 2 and T=9W,=9W 2 =W, n W 2 . Lemma 9.2 Let M be the union of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e and l e t T be an i n c o m p r e s s i b l e t o r u s in M. If T i s nonsepa ra t ing then M i s an o r i e n t a b l e t o rus bundle over S 1 . If T i s s e p a r a t i n g then T decomposes M i n t o W, and W 2 , two o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e . M=wluW2 and T=9W, = 9W2=W1 n W 2 . P r o o f : Two f o l d cover M by a t o rus bundle R. Argue by cases depending on whether p 1 ( T ) i s one or two t o r i . Note the deck t r a n s f o r m a t i o n of M i s a f i x e d p o i n t f r ee i n v o l u t i o n . For d e t a i l s see [11 ] . QED Lemma 9.3 Let M be an o r i e n t a b l e t o rus bundle over S 1 . Suppose M i s a l s o a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e . Then M has a b a s i s as an o r i e n t a b l e t o rus bundle over S 1 so that i t s mat r ix i s £ Q and a ( c a n o n i c a l ) b a s i s as a o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e T 1 a" so tha t i t s mat r ix i s n , . 94 P r o o f : M=T'xI/d fo r some i d e n t i f i c a t i o n d : T ' x - 1 > T x 1 . Let p : T ' x I >M be the induced p r o j e c t i o n . M c o n t a i n s a K l e i n b o t t l e K. I sotope K so tha t p 1 (K ) c o n s i s t s of a n n u l i A^ meet ing both components of 9 ( T ' x I ) . S ince K i s a K l e i n b o t t l e one of the A^ must have boundary components r e p r e s e n t i n g oppos i t e e lements 7 and - 7 of H , ( T ' ) . S ince M i s o r i e n t a b l e i t f o l l o w s M has a mat r ix of the form On the other hand, M i s a union of o r i e n t a b l e tw i s t ed I-bundles W, and W2 over a K l e i n b o t t l e . W, n W 2 = T . T ' de te rmines a nonsepa ra t i ng annulus in each of W, and W 2 . Hence by choos ing a p p r o p r i a t e gene ra to r s we may assume the mat r ix of M, as a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e , i s 1 b' 0 k j . M has a two f o l d c o v e r i n g q:R >M by a o r i e n t a b l e t o rus bundle over S 1 , R = U , U U 2 such tha t U^ = T 2 x l double cove r s W\. The deck t r a n s f o r m a t i o n r e s t r i c t e d to i s a f i x e d p o i n t f r e e i n v o l u t i o n k^  on U^ tha t i n t e r changes boundary components of W\ . Hence k^  i s con juga te to k^ = axKxr. Us ing t h i s i n v o l u t i o n one sees tha t the mat r ix fo r the to rus bundle H i s | g 2^J. The' t o rus T ' which determines nonsepa ra t i ng a n n u l i of W, and W2 must l i f t to two t o r i and t h e r e f o r e a mat r ix fo r R i s a l s o [ -1 a l 2 ["1 - 2 a " |_ 0 - 1 J " L 0 1_ A b e l i a n i z i n g the fundamental group of a t o rus bundle M c 95 wi th mat r ix the mat r ix 1 c . 0 V 1 0' 0 -1 , one sees that H , ( M c ) = Z © Z @ ( Z / c Z ) . A l s o con juga tes [ i to 1 -c 0 1 and M and c M are i s o m o r p h i c , -c I t f o l l o w s b=±a and so b=a fo r a s u i t a b l e c h o i c e of g e n e r a t o r s . QED P r o p o s i t i o n 9.4 Let M be an o r i e n t a b l e t o rus bundle over S 1 or a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e . Let t be an i n v o l u t i o n on M and l e t T be a s e p a r a t i n g i n c o m p r e s s i b l e t o rus w i th T n i T = 0 . Then there i s a nonsepa ra t i ng i n compres s i b l e to rus wi th iT=T and F i x and T t r a n s v e r s a l . P r o o f : By Lemmas 9.1 and 9 .2 , I ^ W ^ L J W V w i th W i n W 2 ' = T where W, and W 2 ' a re o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Without l o s s say iT i s in W 2 ' . Us ing the lemmas aga in we see M = W , u ( T ' x [ - 1 , 1 ] ) U W 2 w i th ( f o r i=1,2) W i n ( T ' x [ - 1 , 1 ] ) = 3Wi = T ' x ( - I ) 1 iW,=W 2 l t ( T ' x [ - 1 , 1 ] ) = T ' x [ - 1 , 1 ] and wi th W i o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Let A be a nonsepa ra t i ng proper annulus in W,. Wr i te 3 A = S 1 U S 2 . There i s an e s s e n t i a l annulus A 0 in T ' x [ 0 , l ] w i th 3A 0=S , u i S 2 . Then 3 A o n i 3 A o = 0 . By the P a r t i a l Annulus Theorem 5.6 there are d i s j o i n t e s s e n t i a l a n n u l i A, and A 2 t r a n s v e r s a l to Fix w i th 9 A o n i 9 A 0 = 9 A 1 U 3 A 2 and wi th e i t h e r i A i = A i ( i=1,2 ) or i A , = A 2 . Let T , = A u t A u A T u A 2 . E s s e n t i a l a n n u l i of T ' X [ - 1 , 1 ] must meet both boundary components so T , i s connec t ed . T , i s a t o r u s . T h i s f o l l o w s s i n ce A i s nonsepa ra t i ng in W, so A i s " t w i s t e d " r e l a t i v e to T ' X [ - 1 , 1 ] as i s tA. i T !=T ! and T , i s t r a n s v e r s a l to F i x . T , i s nonsepa ra t i ng hence i n c o m p r e s s i b l e . QED R e c a l l tha t up to con jugacy there are th ree o r i e n t a t i o n p r e s e r v i n g i n v o l u t i o n s I2C a n d •'o o n a s o ^ i d t o rus V, - f i v e o r i e n t a t i o n p r e s e r v i n g i n v o l u t i o n s k 2 s ' ^40 ' k OF and k^j on a t r i v i a l I-bundle W over a t o r u s and f i v e o r i e n t a t i o n p r e s e r v i n g i n v o l u t i o n s k 2 S ' ' k s ' ' k S 2 c ' ' k 2 c ' and k^' on an o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e . By a p p l y i n g the Torus Theorem 6.2 to M we o b t a i n the f o l l o w i n g theorem. Theorem 9.5 Let M be an o r i e n t a b l e t o rus bundle over S 1 or a un ion of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n bo t t l e ' . Le t i be an i n v o l u t i o n wi th a 1-dimensional f i x e d set component. Then one of the f o l l o w i n g h o l d s : 1) There i s a nonsepa ra t i ng ( i n c o m p r e s s i b l e ) t o rus T w i th i T n T = 0 . T and iT decompose M i n t o two t r i v i a l I-bundles W, and W2 over a t o rus w i th i|W\ con jugate to ^ s ' o r ^01" 2) There i s a nonsepa ra t i ng ( i n c o m p r e s s i b l e ) t o rus T w i th cT=T and i t s c o l l a r i s not i n t e r c h a n g e d . M = W/d where W i s a t r i v i a l I-bundle over a to rus and d i s an isomorphism between the boundary components of W. i i s induced from an i n v o l u t i o n on W that i s con jugate to 3) There i s a s e p a r a t i n g i n compres s i b l e t o r u s T wi th iT=T. M i s the union of o r i e n t a b l e I-bundles W,' and W 2 ' over a K l e i n b o t t l e w i th W 1 ' n W 2 ' = T . t|W,' and i|W 2 ' a re both con juga te to ^520' o r ^2c ' ° r a r e ^oth con juga te to k 2 s ' ' V o r k o ' -4) M s a t i s f i e s case ( I I I ) of Torus Theorem 6 . 2 and F i x ( t ) i s e x a c t l y one 1-sphere. t |V, i s con jugate to J 2 C and i |V 2 i s con juga te to j Q . M i s M 6 . P r o o f : Apply the Torus Theorem 6 . 2 . Assume fo r the moment tha t case ( I I I ) does not o c c u r . Then s i n ce M i s o r i e n t a b l e the re i s a to rus T s a t i s f y i n g case (I) or (I I) of tha t theorem. By P r o p o s i t i o n 9 .4 , i f iT=T then T s e p a r a t e s . I f iT=T and the c o l l a r i s i n t e r changed then we a r r i v e at case (1) by c o n s i d e r i n g the boundary of a c o l l a r of T . We show that case (I I) can be e l i m i n a t e d . Then to complete the p r o o f , case ( I I I ) i s h a n d l e d . Suppose case (I I ) occurs and suppose i S ^ S , . S ince A 1 L j A . 1 i s a s e p a r a t i n g i n c o m p r e s s i b l e t o r u s i t f o l l o w s by Lemmas 9.1 and 9.2 tha t U . , U V , i s an o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e . I A , i s a s e p a r a t i n g annulus so S 1 bounds a proper Mobius band A of U . i . K = A u i A i s a K l e i n b o t t l e w i th iK=K. Then by Lemma 6.1 the boundary of a r e g u l a r ne ighborhood of K i s an i n c o m p r e s s i b l e t o rus i n v a r i a n t under t g i v i n g case ( I ) . Suppose i S ! = S 2 . S ince i has a 1-dimensional f i x e d set and Fixp-15^0, assume t |V•, i s con juga te to j g . As above iA, i s a s e p a r a t i n g annulus in the o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e U . 1 U V , . So S, r e p r e s e n t s twice a genera to r of H , ( V , ) . S e l e c t i n g a p p r o p r i a t e gene ra to r s [M] and [L] of H,(3V 1)=Z@Z where [M] i s t r i v i a l in H , ( V , ) , we may assume [S 1 ]=a[M]+2[L] f o r some i n t ege r a . Le t p r r ^ O V , ) > H 1 ( 9 V 1 / i ) be the obv ious homomorphism. Then p[M] = 2[My] and p[Z-] = [ L 1 ] f o r s u i t a b l e gene r a to r s [A^,] and [L , ] of H , ( 3 V , / i ) . Then p[S,]=2a[Atf 1 ]+2[L,] but t h i s c o n t r a d i c t s that S, >S , / i i s an i somorph ism. 99 Now c o n s i d e r case ( I I I ) . Le t V , " be the c l o s u r e of the component of M - ( T u c T ) tha t c o n t a i n s V , . Let V , 1 be the s o l i d t o r u s ob ta ined from V , " by r e p l a c i n g S by two .copies S, and S 2 . Then V , ' = V , " / ( S y ~ S 2 ) . S i m i l a r l y f o r V 2 ' and V 2 " . Let W, be a r e g u l a r ne ighborhood of T . 9W, i s an i n c o m p r e s s i b l e t o r u s . By Lemmas 9.1 and 9 .2 , W2=M-W, i s an o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e . iT- i n t ( W , ) i s a s e p a r a t i n g a n n u l u s . Let 9(iT- W , ) = C , u C 2 . See F i g u r e 12. It f o l l o w s C 1 f C 2 , S, and S 2 r ep resen t tw ice a genera to r in both H , ( V , ' ) and H , ( V 2 ' ) . Let i , be the i n v o l u t i o n on V , ' induced by i . S i m i l a r l y fo r i 2 . From the Torus Theorem 6 .2 , i , i s con juga te to J 2 Q and t 2 i s con juga te to j g or j ^ . The argument g i ven above fo r case ( I I ) shows t 2 i s not con juga te to jg s i n c e t 2 i n t e r changes components of 9 V 2 - ( S , u S 2 ) . A l s o the 1-ce l l S,/*, in V , ' / i , cannot meet both components of F i x ( t , ) because o therw ise i t s l i f t S! would be a 1-sphere r e p r e s e n t i n g an odd m u l t i p l e of a genera to r of H ^ V , ' ) . Hence S, meets on l y one component of F i x ( t , ) ' . The r e fo r e 1 F i x ( i ) i s one 1-sphere. S ince i s a boundary component of the nonsepa ra t i ng annu lus t T n W , of W, and of the s e p a r a t i n g annulus iT|-|W 2 of W 2 we see tha t a ( 1 ,0 ) -gene ra to r of W, and a (0 ,1 )-genera to r of W 2 c o r r e s p o n d . Let C 2 be a 1-sphere in T meet ing S t r a n s v e r s a l l y in one p o i n t . Let A be the annulus of V 2 ' de te rmined by T . S ince t 2 i s con juga te to j ^ and 3A r e p r e s e n t s twice . a gene ra to r of H , ( V 2 ' ) r the re i s a proper 2 - c e l l D of V 2 ' w i th i2D|-|D=0. A l s o a r range tha t D meets A in two proper nonsepa ra t i ng 1-ce l l s of A, one of which i s C , , and that D meets t 2 A in two proper nonsepa ra t i ng 1 - c e l l s . Then i 2 ( 3 D n i 2 A ) and 3D n A are four d i s j o i n t nonsepa ra t i ng proper 1-ce l l s of A. Now c o n s i d e r A as an annulus in V 2 ' . S ince i , i s con juga te to j 2 c and ( J 2 r j ) * o n H i ^ v ^ ^ s m u l t i p l i c a t i o n by - 1 , i t f o l l o w s tha t i , t 2 ( 3 D n 1 2 A ) u ( 3 D u A ) bounds a d i s c in V 2 . But ( i , i 2 ) | i T = i d so 3D bounds a d i s c D, i n V , ' . By c h o i c e of C 1 f ( D y D , ) n W , i s two Mobius bands so i t s 101 boundary i s a (0 ,1 )-genera to r of" W,. On the o ther hand (D u D 1 ) r - |W 2 i s a nonsepa ra t i ng annulus of W2 s i n c e i t has two boundary components and does not separa te V , ' . T h i s g i v e s M 6 . To show how M 6 a r i s e s in t h i s manner we g i ve the f o l l o w i n g c o n s t r u c t i o n . See F i g u r e 13. Let V j ' = D 2 x S 1 and A j = { ( e 2 7 r i ( t + v ) , e 4 7 r i t ) : teR, 0<V<1/4} Let i , = J Q = i d x a and I 2 = J 2 £ = K X K . Let d be the i d e n t i f i c a t i o n ( e 2 7 r i t e 4 f f i t ) _ ( e 2 7 r i ( l - t ) e 4 * i ( ( 3 / 4 ) - t ) ) on 9A j . Let V j " = V j ' / d . De f i ne h :A 2 >A, to be induced from the i d e n t i t y and d e f i n e h | ( t 2 A 2 ) = i 1 . h . t 2 | A 2 = K X ( - K ) ( i e ) h : 9 V 2 ' >9V,' i s h(x) = x i f xeA, and h(x) = ( K X - K ) ( X ) o t h e r w i s e . Then M 6 = V 1 ' U V 2 ' / h u d and i n v o l u t i o n t , u i 2 has f i x e d set one 1-sphere. (In the p r e v i o u s c o n s t r u c t i o n one can take D u D'={ t=-1/B} ) QED C o r o l l a r y 9.6 Let i be an i n v o l u t i o n on an o r i e n t a b l e t o rus bundle over S 1 or a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e with a 1-dimensional f i x e d s e t . Then the f i x e d set i s one, two, th ree or four 1-spheres. 1 C o r o l l a r y 9.7 Let M an o r i e n t a b l e t o rus bundle over S 1 or a union of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Let t be an i n v o l u t i o n on M wi th a 1-dimensional f i x e d s e t . Then M/c i s a l ens space , P 3, P 3#P 3, or the boundary un ion of a s o l i d t o rus wi th an o r i e n t a b l e t w i s t e d I-bundle over a K l e i n b o t t l e . Proof: Use C o r o l l a r y 7.5 and C o r o l l a r y 8 .5 . Cons ide r the cases of Theorem 9 .5 . In 1) when each of t | i s con jugate to k 2 s or kg , M/t i s a union a long the boundar ies of two s o l i d t o r i . When one of t | i s con juga te to k^T then M/t i s the boundary union of a s o l i d to rus wi th an o r i e n t a b l e tw i s t ed I-bundle- over a K l e i n b o t t l e . In 2) w / k 2 C i s s 2 x I * T n e i d e n t i f i c a t i o n of S 2x-1 wi th S 2X1 g i ves the l ens space S 2 x S 1 . In 3) fo r k 2 s ' , kg ' and k^' we get the same spaces as in 1 ) . For k s 2 c ' and k 2 c ' , c app ing W / k g 2 c ' g i v e s a 3-sphere and capp ing w / k 2 c ' g i v e s P 3. In 4) V / i i s a 3-ce l l and V n V , / t i s two 2 - c e l l s . V , / i i s a 3 - c e l l so VUV,/L i s a s o l i d t o r u s . S ince V 2 i s a l s o a s o l i d t o rus M/i i s a l ens space . QED 103 Note tha t to rus bundles may a l s o be un ions of t w i s t e d I-bundles over a K l e i n b o t t l e . (See Lemma 9.3) In the f o l l o w i n g l e t Q e G L 2 ( Z ) . Let E e G L 2 ( Z ) be i n the i ?] •"<> [I ?]• subgroup K of G L 2 ( Z ) genera ted by Note Z > Z 2 = Z / 2 Z i nduces an exact sequence : 1 >K >GL 2 ( Z ) >GL 2 ( Z 2 ) > 1 Theorem 9.8 Let M be an o r i e n t a b l e t o r u s bundle over S 1 or a union of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Then M has an i n v o l u t i o n f w i th f i x e d set e x a c t l y n 1-spheres (n>0) i f and on l y i f one of a ) , b) or c) below h o l d s . a) M i s a to rus bundle w i th mat r ix con juga te to one of the f o l l o w i n g f o r some Q or E: (1 ) 1 0 0 -1 -1 ( 2 ) Q ' [ (3) Q " 1 [ 1 0' 0 -1 1 -r 0 -1 • [ * - ? ] (4) E P where p = [ rj ? ] (5) E P where p = [ J 1 ( 6 ) E P where p = [ J ? o r [ 1 2 ] n = 2 or 4 n = 1 or 3 n = 2 n = 4 n = 1 or 3 n = 2 104 b) M i s a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e wi th mat r ix (wi th r e spec t to some set of c a n o n i c a l gene ra to r s ) one of the f o l l o w i n g fo r some Q or E: In (7) - (9) the i n ve r s e s of m a t r i c e s are to be taken over the f i e l d of f a t i o n a l s but the product mat r ix i s r e q u i r e d to be i n t e g r a l . (7) (8) (9) (10) EP where P = (11) EP where P = (12) EP where P = (13) EP where P = 1 0" -1 Q 1 0" 0 2_ 0 2 2 0" -1 Q 1 0" 0 1_ 0 2_ 1 0 1 -1 Q 2 0" 0 2_ 0 1_ 1 0 1 1 1 0 Mi !] ]•[! iM? M i J] 2 3 n = 2, 3 or 4 n = 1 or 2 n = 1 or 2 n = 2 or 4 n = 1 or 3 n = 3 ] ~ [-? -! n = 2 c) M i s M 6 and n = 1 Remark 9.9 The proof shows how in each case (1) - (13) the i n v o l u t i o n on M a r i s e s from i n v o l u t i o n s on o r i e n t a b l e t o rus bundles over S 1 or o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Proof: App ly Theorem 9 . 5 . Suppose case (1) of Theorem 9.5 o c c u r s : There i s a nonsepa ra t i ng to rus wi th i T n T = 0 . Let W, and W2 be the two components of M determined by T and i T . Let h,:W, >T 2 xI and h 2 :W 2 >T 2 xI be c o n j u g a t i o n s between i|W^ and the s tandard i n v o l u t i o n s l ^ s ' kg or k Q I w i th h i ( T ) = T 2x1. Let d :T 2x1 >T2x1 be the i d e n t i f i c a t i o n induced by T . Then the i n v o l u t i o n s induce an i d e n t i f i c a t i o n 1 2 . d . t , : T 2 x - 1 >T 2 x-1. Cons ide r the e f f e c t of these isomorphisms in H , ( T 2 ) = H ^ S ' x S 1 ) = Z©Z w i th r e spec t to b a s i s [ S 1 x l ] and [1XS 1 ] where o r i e n t a t i o n s are induced from S 1 c C. Then M = T 2 x l / q where the mat r ix of q i s M ( d ) ~ 1 M ( t 2 ) M ( d ) M ( i , ) . The mat r ix of k 9 C and k i s Suppose case (2) of Theorem 9.5 o c c u r s : There i s a nonsepa ra t i ng to rus wi th iT=T and the c o l l a r of T i s not i n t e r c h a n g e d . Cut M open on T . Then M = T x l / d ' where d ' :Tx-1 >Tx1 i s an i somorph ism, i i s induced from an i n v o l u t i o n X on Tx l that s a t i s f i e s d ' . ( X | T x - 1 ) = ( X | T x 1 ) . d ' . Let h:TxI >T 2xI be a c o n j u g a t i o n between X and k 4 C (with h (Tx-1 )=T 2 x-1 ) . D e f i n e d = h | . d ' . ( h | T 2 x - 1 ) ~ 1 . Then M = T 2 x l / d and i i s con juga te to the i n v o l u t i o n on T 2 x l / d induced from the i n v o l u t i o n k4£=KX/cxid on T 2 x l . The mat r ix of M i s con juga te to the mat r ix of d :T 2 =T 2 x-1 >T 2x1=T 2 . Note d s a t i s f i e s and of k „ i s (1) - (3) above. d . ( K X K ) = ( K X K ) . d . Now T 2 / K X K i s a 2-sphere . The c l a s s of a l l isomorphisms of T 2 / K X K , up to i s o t o p y , tha t keep the four p o i n t s ±1x±1 f i x e d i s genera ted by two Dehn t w i s t s : a t w i s t on S ' x i / / c X K and a tw i s t on i x S V ^ X K , see [ 2 ] , A tw i s t on S'xi/zcx/c l i f t s to a tw i s t on S 1 x i toge the r w i th a tw i s t on S 1 x - i in the "same" d i r e c t i o n . With r espec t to the b a s i s [ S ' x l ] , [ i x S 1 ] of H , ( T 2 ) the l i f t e d tw i s t has mat r ix Hence i f d keeps a l l four p o i n t s f i x e d (and t h e r e f o r e F i x i s four 1-spheres) then the mat r ix of d i s in the subgroup K of G L 2 ( Z ) . In gene ra l d induces a pe rmuta t ion on the four p o i n t s ±1x±1. L abe l these p o i n t s as 1=1x1, 2=-1x1, 3=1x-1 and 4=-1x-1. The number of 1-sphere f i x e d components of t i s the number of o r b i t s of the permuta t ion induced by d on {1 ,2 ,3 ,4 } . Note tha t p ,(z ,w ) = (zw , w ) commutes w i th K X K , induces the permuta t ion (34) and has mat r i x £ Q j j . p, i s in f a c t a Dehn t w i s t . p 2 (z,w)=(-zw,w) has the same mat r ix and induces the pe rmuta t ion (12 ) . p 3(z , w ) = ( z , z w ) commutes w i th K X K , induces the permuta t ion (24) and has ma t r i x | ^ . p „(z ,w )= (z ,-zw) has the same mat r ix and induces the permuta t ion (13) . A l s o note tha t the f o l l o w i n g isomorphisms commute w i th KXK and have mat r ix the i d e n t i t y : ax id i n d u c i n g pe rmuta t ion (12 ) (34 ) , i dxa i nduc i ng (13)(24) and axa i nduc i ng (14 ) (23 ) . 107 There i s a compos i t i on r of these isomorphisms such tha t d = d ' . r and such tha t d ' induces the i d e n t i t y p e r m u t a t i o n : Use a x i d , i dxa and axa to reduce the p o s s i b i l i t i e s to permuta t ions (34 ) , (24) , (14) , ( 12 ) (34 ) , (243) , (234) , (1423) , (1234) and (1243) . Then use the p 's to genera te these pe rmuta t i ons . So the mat r ix of d i s of form EP where E i s the mat r ix of d ' and hence i s in K and P i s of , form l i s t e d in (4) - (6 ) . Suppose case (3) of Theorem 9.5 o c c u r s : There i s a s e p a r a t i n g t o rus wi th iT=T. Then M i s a union of two o r i e n t a b l e t w i s t e d I-bundles W, and W2 over K l e i n b o t t l e s . Let d:3W, >3W2 be the i d e n t i f i c a t i o n . As above, by chang ing t | b y a c o n j u g a t i o n on W^  , we may assume that 11 = 11W, and i 2 =t|W 2 a re s t andard i n v o l u t i o n s . Then i 2 d = d i 1 . S e l e c t r e p r e s e n t a t i v e s of H,(3W 1)=Z@Z in the c a n o n i c a l way: (1,0) a r i s i n g from a nonsepa ra t i ng annulus of W, and (0,1) from a s e p a r a t i n g annu lu s . Suppose t 19W! i s f i x e d p o i n t f r e e . T ^ S V ^ / i , i s a t o r u s . We can s e l e c t a b a s i s f o r T , such tha t p,:3W, >T, i f 1, i s con juga te to k 2 s ' o r ^s' ,„[ T 2 . d:3W, >3W2 induces an isomorphism q : T , >T 2 w i th p 2 „ d = q . p 1 . Then the mat r ix of d i s (as a p roduct in GL 2 (Q ) ) the mat r ix M(p 2 )~ 1 QM(p , ) where Q i s the mat r i x of q . has mat r ix M(p,)= and mat r ix | Q ^ 108 C o n v e r s e l y , up t o i s o t o p y a m a t r i x Q e G L 2 ( Z ) d e t e r m i n e s an i s o m o r p h i s m q:T, >T 2 and t h i s i s o m o r p h i s m l i f t s i f M ( p 2 ) ~ 1 Q M ( p , ) e G L 2 ( Z ) . T h i s g i v e s (7) - ( 9 ) . Suppose t|9W, i s n o t f i x e d p o i n t f r e e . Then t|9W, i s c o n j u g a t e t o KXK. I , and t 2 a r e c o n j u g a t e t o k 2 £ ' = ( - / c j x r x r / d o r t o k S 2 £ ' = i d x r x T / d . F o r c o n v e n i e n c e use t h e r e p r e s e n t a t i o n K x r x i d / d f o r k g 2 ^ ' i n s t e a d . F o r b o t h k 2 c ' and k s 2^,' : S ^ S ' x O x l i s i n v a r i a n t , meets b o t h f i x e d 1 - c e l l s and r e p r e s e n t s (1,0)eH,(9W,), t h a t i s , i s a b o u n d a r y component of a n o n s e p a r a t i n g a n n u l u s . F o r ^ 2 c ' : S 2 = 1x1x1 JJ 1x-1x1 i s i n v a r i a n t , meets b o t h f i x e d 1 - c e l l s and r e p r e s e n t s ( 0 , 1 ) . F o r k ^ ^ ' : S 2 = i x 1 x i u - i x - 1 x i i s i n v a r i a n t , meets o n l y one f i x e d 1 - c e l l and r e p r e s e n t s ( 0 , 1 ) . The c u r v e s S, and S 2 g i v e a way t o a s s i g n l a b e l s 1, 2, 3 and 4 t o t h e f o u r p o i n t s of F i x n 9 W , ( e . g . 1 t o S i n S 2 ) . The f i x e d s e t s o f k 2 ^ ' and kg2c' m a t c n t h e s e l a b e l s i n a d i f f e r e n t way. The m a t c h i n g c a n be a r r a n g e d t o o c c u r i n t h e same way i f a t w i s t on S 2 i s p e r f o r m e d i n t h e k g 2 c ' c a s e . The m a t c h i n g and d d e t e r m i n e t h e number of components of F i x . P r o c e e d as i n t h e c a s e above where T was a n o n s e p a r a t i n g t o r u s w i t h iT=T. In t h e above l i s t i n g (12) and (13) a r i s e f r o m c o m b i n i n g a k 2 c ' and a k g 2 c ' . (11) and (12) a r i s e f r o m c o m b i n i n g a k 2 (,' and a k ^ ' o r from c o m b i n i n g a k S 2 c ' a n d 3 k S 2 c ' QED 1 Let M be an orientable torus bundle over S'1 with involution i . C a l l t f i b e r p r e s e r v i n g i f there i s a f i b r a t i o n p:M >S1 such that i(p 1(x))=p 1 ( i ( x ) ) for a l l xeS 1 and i f Fix=Fix ( i ) is transversal to each fiber p 1 ( x ) . Note the involution t(x,y,z)=(y,x,-z) on S^S'xS 1 i s not fiber preserving, in t h i s sense, with respect to f i b r a t i o n obtained by projection to the l a s t coordinate. Corollary 9.10 Let M be an orientable torus bundle over S 1. Then M has a fiber preserving involution with fixed set exactly n 1-spheres (n>0) i f and only i f the matrix of M i s conjugate to one of (4), (5) or (6) of Theorem 9.8. Proo f : Let xeS 1 be such that the fiber T=p 1(x) meets the fixed set. Then cT=T. Since T and F i x are transversal- the c o l l a r of T is not interchanged. So T s a t i s f i e s case (2) of Theorem 9.5. Now follow the proof of Theorem 9.8. QED 1 10 #10. I n v o l u t i o n s With 2-Dimensional F i x e d Sets In the p r ev i ous s e c t i o n the space forms M, , M 2 and M 6 were d e f i n e d . R e c a l l a l s o D e f i n i t i o n 3 .1 . De f i ne the f o l l o w i n g i n v o l u t i o n s on these spaces . On M, : a 2 T = i d x K x i d / d hav ing f i x e d set two t o r i S 1 x ± 1 x l / d a T=a)xid/d hav ing f i x e d set the nonseparat i ng- to rus { ( z , z ) | zeC }x I /d On M 2 : /3 2 K=idx/cxid/d hav ing f i x e d set two K l e i n b o t t l e s S ' x i l x l / d /3K=wxid/d hav ing f i x e d set the K l e i n b o t t l e {(z , z) | zeC}xI/d / 3 T = i d x ( - K ) x i d / d hav ing f i x e d set s e p a r a t i n g t o r u s S ' x l i x l / d /3 T 4 p=KX/cxr/d hav ing f i x e d set a t o r u s and four p o i n t s ( S ' x S ' x - l u ± 1 x ± 1 x 0 ) / d On M 6 : 7 K 2 p = ( - K > x a x i d / d hav ing f i x e d set a K l e i n b o t t l e and two p o i n t s (S 1 xS 1 x1 u 1 x ± 1 x - 1 ) / d Here d denotes the i d e n t i f i c a t i o n fo r the c o r r e s p o n d i n g space form M. . 111 Lemma 10.1 Let M be an o r i e n t a b l e t o rus bundle over S 1 or a union of o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . Let i be an i n v o l u t i o n w i th f i x e d set c o n t a i n i n g a K l e i n b o t t l e . Then the re i s a s e p a r a t i n g i n c o m p r e s s i b l e t o r u s T w i th iT=T and T n F i x = 0 . Proof: By Lemma 6.1 the boundary of a r e g u l a r ne ighborhood of the K l e i n b o t t l e K i s an i n c o m p r e s s i b l e t o r u s . S ince K i s f i x e d under t, a r range tha t the r e g u l a r ne ighborhood i s i n v a r i a n t under t and tha t i s meets F i x on l y at K. Then l e t T be the boundary of t h i s r e g u l a r ne ighborhood . QED Theorem 10.2 Let M be an o r i e n t a b l e t o rus bundle over S 1 or a union of o r i e n t a b l e tw i s t ed I-bundles over a K l e i n b o t t l e . Let t be an i n v o l u t i o n on M wi th a 2-d imens iona l f i x e d set component. Then M i s i somorph ic to M l f M 2 or M 6 and i i s con juga te to one of the seven i n v o l u t i o n s d e f i n e d above. P r o o f : We app l y the Torus Theorem 6 . 2 . Some economy c o u l d be ache i ved by showing that the f i x e d set c o n t a i n s an i n c o m p r e s s i b l e t o r u s or K l e i n b o t t l e F. When F i s a t o r u s , F c o u l d then be used to c o n s t r u c t a 1 12 t o r u s T with i T n T = 0 . The present approach has the advantage that i t g e n e r a l i z e d to the case of o r i e n t a t i o n r e v e r s i n g i n v o l u t i o n s . I t a l s o p a r a l l e l s the proof of Theorem 7.3. M i s o r i e n t a b l e so case (IV) of the Torus Theorem 6.2 does not a r i s e and 2-sided K l e i n b o t t l e s do not occur. Cases (II ) and ( I I I ) do not a r i s e s i n c e t i s o r i e n t a t i o n r e v e r s i n g . So there i s an incompressible t o r u s T with e i t h e r T n i T = 0 or iT=T and T and F i x t r a n s v e r s a l . By P r o p o s i t i o n 9.4 there are three c a s e s : Case 1) i T n T = 0 and T does not separate. Case 2) iT=T, T does not separate and the b i c o l l a r of T i s not interchanged. Case 3) iT=T and T s e p a r a t e s . By the l a s t lemma we may assume T does not meet any K l e i n b o t t l e components of F i x . We show t i s conjugate t o : i n case 1) a 2 T , o T , or / 3 T 4 p . i n case 2) a T i f T n F i x i s one 1-sphere, a 2 T , a T , or 0 T otherwise, i n case 3) 0 2 R , 0 R , 0 T, / 3 T 4 p or 7 K 2 p -S e v e r a l of the standard i n v o l u t i o n s are l i s t e d i n more than one case. Each standard i n v o l u t i o n can a r i s e i n the case i t i s l i s t e d under. Namely c o n s i d e r T=: S ' x i x l , { ( x , i x , t ) : x , t } , S 1 X S 1 X ( 1 / 2 ) , S'xS'xO, S'xS'xO, 1 1 3 { ( x , x , t ) : x , t } , S 1 x S 1 x O , S ' x t i x l , { ( x , ± i x , t ) ; x , t } , t i x S ' x I , S 1 x ± i x l and S 1 x S 1 x O , r e s p e c t i v e l y . I t s u f f i c e s to show t has the same f i x e d set as a s tanda rd i n v o l u t i o n l i s t e d under the c o r r e s p o n d i n g c a s e , and tha t i f L and i ' have isomorphic f i x e d se t s and f a l l i n t o the same case 1)- 3) then they are c o n j u g a t e . We make use of Lemmas 9.1 and 9 .2 , C o r o l l a r y 7.4 and C o r o l l a r y 8 .4 . Case 1) i T n T = 0 and T does not s e p a r a t e . i T u T decomposes M i n t o two components W^  = T 2 x l w i th 3W^=T U i T . S ince the re i s a f i x e d s e t , tW^=W .^ The boundary components of W^  a re in te r changed so t|(W^) i s con juga te to k T , k 4 p or k N I . S ince t has a 2-d imens iona l f i x e d set assume t|(W,) i s con juga te to k T . F i x e d se t s of the same type as fo r the s tandard i n v o l u t i o n s C ^ T ' a T ' o r ^T4P a r e o b t a i n e d . G iven t' w i th i somorph ic f i x e d s e t , l e t h:W2 >W2' be a c o n j u g a t i o n between t|(W 2) and i | ( W 2 ' ) . Then ( h p t B W ^ a W j >3W2'=9W1I i s a c o n j u g a t i o n which by the c o n j u g a t i o n ex tendab le p rope r t y f o r k T extends to a c o n j u g a t i o n h:W, >W,' . Hence i and t' a re c o n j u g a t e . Case 2) iT=T, T does not separa te and t does not i n t e r change the c o l l a r . Then M i s i somorph ic to W/d where W=T2xI and d :T 2 x1 >T 2x-1 i s an isomorph ism. t i s induced by an i n v o l u t i o n X on W tha t does not in te r change the components of 9W. Since X has a two dimensional fixed component, X is conjugate to k A or k 2 A« Case 2 . 1 ) X is conjugate to k A > Then 9 F i x ( X ) = S , u S 2 has two components so Tr-|Fix has one component. Orient S,. Then annulus F i x ( X ) induces an orientation on S 2. If d*[S,]=-[S 2J then F i x is a Klein bottle meeting T. So d*[S 1]=[S 2 3 and F i x i s a torus. Let i ' be conjugate to t and assume i ' f a l l s into t h i s case also. Construct W/6" = M' as above. Construct a conjugation T 2 x1 > ( T 2 X 1 ) ' between X | ( T 2 X 1 ) and X ' | ( T 2 x 1 ) ' . Extend to a conjugation h:9W >9W by defining h(T 2x-1)=d'.h.(d 1 ) . Since F i x i s a torus in an orientable manifold, h extends over a b i c o l l a r of F i x . Then the conjugation extendable property of k A shows h extends to a conjugation h:W >W between X and X'. h induces a conjugation between t and i ' Case 2 .2 ) X is conjugate to k 2 A« Then F i x ( X ) has two annular components A , and A 2 and F i x meets T in two 1-spheres. Let S ^ j = T 2 x ( - 1 ) 1 n A j . Pick orientations so that each represents the same element of H,(W). d:T 2x1 >T2x-1 i s orientation preserving and must take S,, to S 2 1 or S 2 2. There are four subcases: 1) d(S 1 1)=S 2, and d*[S,,] = [S 2 , ]. Then F i x i s two t o r i . 2) d(S, 1)=S 2 1 and d*[S,,]=-[S 2 , ] . Then F i x is two Klein b o t t l e s . These meet T so th i s case does not occur. 3) d ( S l 1 ) = S 2 2 and d*[S,,]=[S 2 2]. Then F i x i s a 1 15 nonsepa ra t i ng to rus s i n c e d must i n te r change the components of W-Fix. 4) d ( S 1 1 ) = S 2 2 and d * [ S , , ] = - [ S 2 2 ] . Then F i x i s one s e p a r a t i n g t o r u s . If i ' i s a l s o g i ven then i t w i l l f a l l i n t o the same one of these subcases . So a c o n j u g a t i o n can be c o n s t r u c t e d between X and X' as in Case 2 . 1 . Case 3) iT=T, T s e p a r a t e s . Then M=W, U W 2 w i th T=W, n W 2 where W\ are o r i e n t a b l e t w i s t e d I-bundles over a K l e i n b o t t l e . We have tW,=W, and i|(W^) i s o r i e n t a t i o n r e v e r s i n g . So X i = t | W i i s con jugate to k R , k 2 M , k A , k A 2 P ' k 2 p or k^. S ince i l a w ^ i l S W j i t f o l l o w s : Case 3.1) Both X, and X 2 a re con juga te to k R , k 2 p or k N Case 3.2) Both X, and X 2 a re con juga te to k 2 M , k A or k A 2 p . In case 3.1) by symmetry assume X, i s con juga te to k R . F i x i s two K l e i n b o t t l e s , a K l e i n b o t t l e and two p o i n t s , or j u s t one K l e i n b o t t l e . I f i ' i s a l s o g i ven c o n s t r u c t a c o n j u g a t i o n by t a k i n g any c o n j u g a t i o n between X 2 and X 2 ' and ex tend ing to W, us ing the c o n j u g a t i o n ex tendab le p r o p e r t y of k K . In case 3.2) the f i x e d set always i n t e r s e c t s T . By c o n s t r u c t i o n we avo ided K l e i n b o t t l e s meet ing T . T h e r e f o r e X^ i s not con jugate to k 2 M < Next , i t i s p o s s i b l e tha t X, i s con jugate to k A and X 2 i s con juga te to k A 2 p . Let F i x n T = S 1 U S 2 . Torus T induces an o r i e n t a t i o n on S 2 once an o r i e n t a t i o n on S, i s f i x e d . In k. the annu la r f i x e d set induces the same o r i e n t a t i o n on S 2 as T does . In t ' i e a n r > u l a r f i x e d set induces an o r i e n t a t i o n on S 2 o p p o s i t e to the one T i n d u c e s . T h e r e f o r e T would be a K l e i n b o t t l e . Hence we have X, and X 2 are con juga te to k A and the f i x e d set i s a t o rus or X, and X 2 are con juga te to k ^ p a n ^ the f i x e d set i s a t o rus and four p o i n t s . I f i ' i s g i ven i t i s easy to c o n s t r u c t a c o n j u g a t i o n between i ' and i s i n c e k A and k A2p n a v e t n e c o n j u g a t i o n ex tendab le p r o p e r t y . QED 1 17 B i b l i o g r a p h y [ I ] R. H. B i n g , A homeomorphi sm between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. 56 (1952) , 354-362. [2] M. Dehn, Die Gruppe der Abbi I dungsklassen, Ac ta Math. 69 (1938) , 135-206. [3] A. C . Gordon and R. A. L i t h e r l a n d , Incompressible surfaces in branched coverings, p r e p r i n t . [4] W. Hantzsche and H. Wendt, Drei dimensi onale euklidische Raumformen, Math. Ann. 110 (1935) , 593-61 1 . [5] J . Hempel, 3-manifolds, Ann. of Math. S t ud i e s 86, P r i n c e t o n Un i v . P r e s s , 1976 [6] P. K. Kim and D. E. Sanderson , Orientation-rever sing PL involutions on orientable torus bundles over S 1 , Mich igan Math. J . , 29 (1982) , 101-110. [7] P. K. Kim and J . T o l l e f s o n , PL involutions of fibered 3-manifolds, T r a n s . Amer. Math. Soc . 232 (1977) , 221-237. [8] K. Kwun and J . T o l l e f s o n , PL involutions of S ' x S ^ S 1 , T r a n s . Amer. Math. Soc . 203 (1975), 97-106 [9] G. R. L i v e s a y , Involutions with two fixed points on the three-sphere, Ann. of Math. 78 (1963) , 582-593. [10] E. L u f t , Equivariant surgery on incompressible annuli and tori with respect lo involutions, to appear . [ I I ] E. L u f t and D. S j e r v e , Involutions with isolated fixed points on orientable flat 3-di me ns i o nal space forms, to appear in T r a n s . Amer. Math. Soc . [12] P. O r l i k , Seifert manifolds, L ec tu re Notes in Math. 291, Sp r inge r V e r l a g , 1972. [13] C. Rourke and B. Sanderson , Introduction to Pi ecewi se-Li near Topology, E r g . der Math. u. i h r e r Grenz . 69, Sp r i nge r V e r l a g , 1972. [14] J . T o l l e f s o n , Involutions of sufficiently large 3-mani folds, Topology 20 (1981) , 323-352. 118 [15] J . Wol f , Spaces of constant curvature, McGraw-H i l l , 1967. [16] F. Waldhausen, Vber Involutionen der 3-Sph'dre, Topology 8 (1969) , 81-91. 

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