INVOLUTIONS WITH 1- OR ORIENTABLE TORUS ORIENTABLE 2~DIMENS IONAL BUNDLES FIXED POINT OVER A 1-SPHERE AND SETS ON UNIONS OF TWISTED I-BUNDLES OVER A K L E I N BOTTLE by WOLFGANG HERBERT HOLZMANN B.A., University A THESIS SUBMITTED IN of Calgary, PARTIAL 1976 FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We a c c e p t to this the thesis required as c o n f o r m i n g standard THE UNIVERSITY OF BRITISH COLUMBIA March 1984 © Wolfgang Herbert H o l z m a n n , 1984 ON OF In presenting this thesis in partial fulfilment of r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t of B r i t i s h it freely agree for Columbia, available I agree the University that the L i b r a r y for reference and s t u d y . that permission for extensive copying of understood that financial copying or p u b l i c a t i o n of I make further this thesis Department o f Mathematics The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 April this It is thesis g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n permission. Date shall s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . for the 23 r d r 19B4 Columbia Thesis Supervisor; Dr. Erhard Luft Abstract We obtain involutions orientable a complete on 3 - m a n i f o l d s nor i s H,(M) p r o c e e d s by a s u r g e r y Similar These are over a have S Klein applied to four the 1-spheres. these spaces orientable twisted the orientable has two a unique set flat over S a Klein 3-space involutions, 1 M 2 has theorem fixed point sets. It are two, determine a given is three, or which of number of obtained. fixed point and on u n i o n s of bottle are M,, fixed sets. four is spaces with forms I-bundles these with 2-dimensional with 2-dimensional involution. torus torus of c o n d i t i o n s that bundles over spaces bottle. twisted must be o n e , sets discs. orientable d e t e r m i n i n g which fixed point I-bundles involutions M, torus be The p r o o f various a Klein to equivariant involutions involutions for to and f o r on orientable 1-dimensional Matrix V - s p h e r e s as t h e The The fixed point have required annuli attention u n i o n s of with for I-bundles over t h e p r o b l e m of that not theorem t o be i n f i n i t e . given our or 1 is involutions twisted bottle. involutions shown are we r e s t r i c t over M torus argument. used to c l a s s i f y Next bundles M. restricted theorems s u c h as o r i e n t a b l e equivariant sets orientable classified. M 2 and Up t o involutions, on M Only have 6 conjugacy, and M 6 has iii Table of C o n t e n t s Abstract List of ii figures v Acknowledgement vi Introduction 1 Chapter I. Equivariant Transversality and D i s c T h e o r e m s . . . . <|>1. P r e l i m i n a r i e s §2. II. III. Involutions 4 Transversality 8 on t h e 3 - C e l l and t h e S o l i d T o r u s . . . 25 #3. Some I n v o l u t i o n s 25 #4. Involutions 31 Equivariant #5. IV. Equivariant 4 A n n u l u s and T o r u s Theorems 42 Equivariant 48 Tori T o r u s Theorem on O r i e n t a b l e I - B u n d l e s O v e r and K l e i n Involutions Bottles on t h e T r i v i a l 62 I-Bundle Over a T o r u s #8. 42 A n n u l u s Theorems Involutions #7. on t h e S o l i d T o r u s Involutions Over a K l e i n 62 on t h e O r i e n t a b l e Bottle I-Bundle 76 i v V. Involutions on O r i e n t a b l e T o r u s B u n d l e s O v e r a 1-Sphere and on U n i o n s of O r i e n t a b l e I - B u n d l e s Over Klein Twisted Bottles 90 #9. Involutions With 1-Dimensional F i x e d Sets . . 90 #10. Involutions With 2-Dimensional F i x e d Sets . 110 B i b l iography 117 List of Figures Figure 1 6 Figure 2 Figure 3 24 Figure 4 27 Figure 5 32 Figure 6 43 Figure 7 50 Figure 8 53 Figure 9 55 Figure 10 63 Figure 11 78 Figure 12 99 Figure 13 100 •. . 9 Acknowledgement I wish to thank my s u p e r v i s o r , guidance and e n c o u r a g e m e n t I e x p r e s s my g r a t i t u d e wish to Engineering British Research Columbia for I appreciate in various project. ways With have been I given were wish to to C o u n c i l of the the efforts Natural Sinan Luft, t o me.. A l s o Sertoz's the financial of support. persons completion helpful and University numerous o t h e r to whose Sciences Canada and t h e of contributed Erhard invaluable p r o v i d i n g me w i t h numerous p r o b l e m s e n c o u n t e r e d computer Dr. of who this suggestions in producing t h i s thesis the on a overcome. thank me i n many my p a r e n t s ways. for the help that they have INTRODUCTION We with investigate t h e p r o b l e m of 1-dimensional or torus bundles over I-bundles over a incompressible orientable In S tori Chapter bottle. T gives be done i n a manner t h a t the f i x e d set Fix, one of tori i.e., are classifying or equivariant involutions twisted that the theorem Klein bottle. exceptional To is a extends these may have t o a l l o w the Even cases are prove techniques are the used. spaces tori = 0 or and the of In first torus the = reduces to over bottle. but under homology b e i n g isolated the iT a Klein [14] to i and I-bundles In was p r o v e d under t h e results. in n orientable. number cutting iT T I-bundles over as, [11] an assumption points. Our n o n o r i e n t a b l e case to be orientable or torus involution trivial [8] on the c u t t i n g The p r o b l e m t h e n being theorem f i x e d set either on such manifold torus twisted bottle. r e s p e c t s both the such t h a t hypotheses, and t h e orientable I-bundles over a Klein Such t h e o r e m s have been p r o v e d i n infinite these T h i s theorem a l l o w s transversal. orientable additional on we p r o v e a c o m p l e t e e q u i v a r i a n t involutions. T and T and F i x trivial involutions orientable Cutting I-bundles over III fixed sets or on u n i o n s of 1 Klein twisted theorem f o r 2-dimensional classifying replaced case we by a two t y p e s of possible. equivariant torus An e q u i v a r i a n t theorem cut transversality and paste theorem is also required. In the two d i m e n s i o n a l c a s e nonorientable "transversality interesting exceptional saddle p o i n t s . the surgery Saddle to involutions the orientable the torus on t h e I-bundle over M denote In Kim and S a n d e r s o n Our orientable techniques allow us t o spaces are the orientable see [15]. M finite. We show having involutions is involutions far has case less of *)9 are called in with are not which the classified set flat 3-space not a 1-dimensional M have S on or 1 a Klein a bottle. orientation S . 1 these forms M,, M ,••'M , 2 6 b u n d l e and H , ( M ) 6 6 are the fixed only point is M sets. conjugacy. on t h e s e orientable fixed involutions with space point o n l y M w i t h such i n v o l u t i o n s . spaces and bundles over involutions the a torus M. A s u b c l a s s of and M d e t e r m i n e d up t o of classify with torus 2 to disc involutions 2-dimensional 1-dimensional with the on a l l M IV the torus classify Each a n n u l u s and I-bundles over £ 1 0 that M,, restrictive. involutions these in in is 6 the bundle over on o r i e n t a b l e fixed point The be bottle. torus have 2-dimensional These Certain separately I-bundle over a Klein twisted involutions Wolf will used in Chapter an o r i e n t a b l e of reversing these theorem a r e trivial union [6] arise; p o i n t s must be t r e a t e d These theorems are the Let points be g u a r a n t e e d . arguments. Analogous theorems. c a n not when M i s We spaces forms sets but determine 1-dimensional fixed point uniqueness sets. in Several For this topics example, torus surfaces the results actions of do not for S . 1 higher further the with the problem research present involutions on t h e Can an e q u i v a r i a n t genus? would g e n e r a l i z e and f i n i t e deal of thesis. classify bundles over for We group a c t i o n s . nonorientable t h e o r e m be One c o u l d a l s o from themselves. proved investigate involutions to how n-cyclic I. EQUIVARIANT TRANSVERSALITY AND DISC #1. THEOREMS Preliminaries Use subset. to , n |_j and u does not indicate to denote set intersection, disjoint union. union Use upper and indices dimension. Throughout is to denote c avoid topological we use wild fixed category, A piecewise the see linear piecewise sets linear which can category. arise This in the MJ. homeomorphism will be called an 9M and F a is proper if isomorphism. Def i n i t i o n Let 1.1 M be a manifold submanifold of M of F 3M=3F. particular In n in the are proper. F interior will usually A surface F a 2-sphere not lower a point denote or M is each Let i is an proper in M only assume t h a t a surface, 2-cell is if is F a 2-cell and i f D 2-sphere M be a c o n n e c t e d compact isomorphism with i * i d and c all for F with if incompressible each 2-cell 3D=3B. A 3-manifold. i =id. is connected An if B in M manifold i n M bounds a 3 - c e l l 2 it submanifolds a compact F -in a 3 - m a n i f o l d M i s w i t h B |-| F = 3B t h e r e irreducible boundary dimension. of M. We w i l l manifold. is with in M. involution Let F i x denote i be an i n v o l u t i o n manifold M'. isomorphism is L 1 and = h 1 are >M' w i t h an i n v o l u t i o n if conjugate i'=h.i.h 1 . Let Call there h a on a is an conjugation t'. and ' I9M' °" i' conjugation i s e t F i x = f i x ( c ) = {x : i ( x ) = x } . on a m a n i f o l d M and i ' h:M i and between of i the f i x e d an I9M ° c h if extendable given isomorphism 1 t n e there n any c o n j u g a t e h :9M >9M' 0 t' with i s a c o n j u g a t i o n h:M >M' e x t e n d i n g the isomorphism h . 0 Note t h a t conjugate suffices with c'. t i s conjugation extendable Further, to check respect depending for if the case to on a the any h e H w i t h 0 of 1 0 extendability ) extendable isomorphisms i' there of is t, a it 9M if >9M', at least conjugation >M' e x t e n d i n g t h e i s o m o r p h i s m h . 0 Figure 1. and B ' . h |:B iB 0 i' |B' we H class so i s any i is conjugation c h o i c e of c o n j u g a t e The f o l l o w i n g M' i ' = t. i' | = h • i | • ( h 0 h:M t o show c o n j u g a t i o n then n n i'B'. define conjugation Let construction M=B iB u Let h :B 0 >B' t'B' n h 0 with B >B' is will iB= 9B n be be n an used between 0 and isomorphism a conjugation hi between t|M and t' _ = i ' . h « ( i M. 0 JM'. See 9 iB and s i m i l a r l y i s e x t e n d e d by e q u i v a r i a n c e h|_=h often. for such that i|B iB t o h:M Then n >M' h is and if a 6 t Figure Lemma 1.2 Given t of a simplicial M there respect to Let a s u b d i v i s i o n L of t be an involution of M w i t h barycentric disjoint 0-, 0 0-, v. is veFix v. In If LJFIX 1 then if then respectively t is M is 0 if Fix 1 * 0. L' be the L'. Fix is the union submanifolds. for the u n i o n s of the Fix. 2 veFix if and l e t L be a 2 and F i x particular reversing with L. a s u b c o m p l e x of 1 , Fix >L s i m p l i c i a l 1- and 2 - d i m e n s i o n a l p r o p e r 0 at i:L 1- and 2 - d i m e n s i o n a l c o m p o n e n t s of If t simplicial on a m a n i f o l d M. L e t s u b d i v i s i o n of Then F i x = f i x ( t ) Write Fix K with involution 1.3 subdivision first is s u b d i v i s i o n K of M and an L. P r o p o s i t ion of 1. Fix t is locally locally orientable orientation orientation then t reversing preserving is at orientation 2 jjFix * 0 and t is orientation preserving 7 Proof: Use the 1) Let A following: be a standard subdivision) invariant of barycentric the first 2) If If veFix 3) If under i. m-simplex Then F i x A a vertex s u b d i v i s i o n of L&nFix of then int(L) contains a is n F i x c o n t a i n s a 3-simplex is (with a standard subcomplex A. i= id. c o n s i d e r the 1-cell then link Lk of v. is one follows m=2 Lk Fix n 2 1-sphere. 4) If So v e F i x . L / c r - | F i x c o n s i s t s of m>0 v e r t i c e s - m = i xUk/i) S i n c e Lk/i is a surface and h e n c e v e F i x . , 1 or it and Lk follows - m) (\(Lk) is then a 2-sphere it m=0 and hence veFix**. QED Corollary Let M/t 1.4 be an = M/(m~i(m) for t singularities. i n v o l u t i o n on M w i t h f i x e d s e t all Fix s u b m a n i f o l d s Fix**, F i x 1 meM) is = Fix/i and F i x a 2 manifold a Fix. with disjoint w i t h Fix** and Fix Then possible union 1 of proper 2 i n M/t and w i t h F i x Proof: C o n s i d e r the 2 a s u b m a n i f o l d of link Lk of 3(M/i) vertices of = (9M)/i Fix u . Fix. QED Remark 1.5 The fixed correspond to then involution 1 (x) if p:M §2. coverings In F 0 order in a isotopy property by 0 that on is manifold M an involution If i is 2 - f o l d c o v e r e d by M. cover interchange is a the then the define an two p o i n t s nontrivial of deck p. M fixed set Fix b r a n c h e d on F i x . t| correspond M-Fix is to fixed i s unbranched. Transversality t o be a b l e 3-manifold on F t with and p| M - F i x Equivariant a 2-fold xeX. on M meM) t to i n d u c e d by Involutions free involutions all >X i s every transformation point for i by r e q u i r i n g for 2-fold free 2 - f o l d c o v e r i n g s by M. M/i=M/(m~c(m) Conversely, p point M we w o u l d l i k e such t h a t F, to perform surgeries the a surface t o p e r f o r m an a m b i e n t isotopic i F and F i x a r e on surface pairwise F has the transversal. This 2 c a n be done i f is the m a n i f o l d i s nonorientable, possible weaker in g e n e r a l . f o r m of however This orientable. pairwise F i x *0 and transversality necessitates transversality. If using a is M not somewhat Lemma 2.1 Let and are F n 1-spheres n The the 2 a point i:B in C F i x >B 3 intersection 1-sphere Fix statement 2 n the 3-cell L If C is a follows n follows of of F on c o n s i d e r i n g t h e star . QED B ={(x,y,z): 3 of B 3 as |x | < 1 , | y | < 1 , | z | < 1 } with the y z the j o i n of plane. Let in take D d e f i n e d z y by S be {(1 , 1 , 1 ) , ( - 1 , - 1 , 1 )} s h a p e d r e g i o n . See F i g u r e saddle i F by t r a n s v e r s a l i t y S. alternately F 2 . and l e t D be t h e cone a iF R be t h e map i ( x , y , z ) = ( - x , y , z ) . Then F i x ( i ) obtained is F, component {(-1 ,1 ,-1) , ( 1 f - 1 , - 1 ) } D with Then t h e components o f 1-cells. *0 t h e n C first 3 in a 3-manifold transversal. i F . The s e c o n d s t a t e m e n t For let surface and p r o p e r with C F i x Proof: of pairwise Fix cF and F be a p r o p e r from ( 0 , 0 , 0 ) 2. 3 is the with on (We c o u l d {z=xy/Vx•x+y-y} {(0,0,0)}.) u 10 Notice that D iD while D Fix(i) are not u part part transversal at of of and i D and F i x ( i ) making t h e s e 3 the is n transversal 3B is n the the are and y axis in B y a x i s . D and F i x ( i ) transversal, (0,0,0). spaces s i m p l i c i a l x There with all are but D and is 3 iD a subdivision the vertices on (0,0,0). Definition Let 2.2 F be a p r o p e r involution surface in a 3-manifold M on M w i t h f i x e d s e t Fix. Call and a point i v a an saddle 2 point to if veF Fix and i f n (F, i F , F i x ) s t a r ( v ) is n isomorphic (D,iD,Fix(i)). Remark 2.3 Saddle fixed set transversal Fix(i) Fix(i) points exist is an involution Although 3D, 9iD, 9Fix(i) are there no 2 - c e l l E with 9E=9D and is transversal. (±1,0,0) (1,0,0)e9l i Fix(i). pairwise = since and t h i s there is Otherwise, a 1-cell 1 - c e l l must meet F i x , since I of with pairwise E, iE, 9E 9iE n E n i E - with contradicting the p r e v i o u s lemma. Let all B /d 3 iF, d denote the identification x and z . Then D/d i s and no an a n n u l u s i s o t o p y of D/d moves and F i x ( i ) / d p a i r w i s e (x,1,z)~(x,-1,-z) in a s o l i d K l e i n it transversal. for bottle t o an a n n u l u s w i t h F, 11 Definition Let 2.4 F involution almost 1) be a surface on M w i t h f i x e d pairwise F, proper transversal cF and F i x a r e finite number of and 2) The points are saddle only F i x . Then F, cF, and t an and F i x are if: pairwise transversal except at a points, components 1-spheres one s a d d l e set in a 3-manifold of and e a c h F Fix containing n such 1-sphere saddle contains at most point. Let E be the disjoint 1-spheres closure of (F iF) - Fix . n and p r o p e r 1-cells: E consists in a neighborhood of of 2 a saddle point, [-1,0)xOxO (0,1]xOxO in u Let v. F n i F the B E be a component of Then E has a fixed Therefore, either than is v or E a E point is a 1-sphere 3 Fix model corresponds for saddle E that contains and is with e x a c t l y two case By w transversality is proposition: a saddle w is in F i x point. We fixed point under, 1 - c e l l w i t h no f i x e d p o i n t s 1 and w. points. a saddle invariant to i. other points v 2 or F i x obtain . In the the latter following 12 P r o p o s i t i o n 2.5 Let the 1) LF and F i x be a l m o s t F, components o f F n i F pairwise transversal. a r e o f one o f t h e f o l l o w i n g Components w i t h no s a d d l e p o i n t s forms: (standard components): 2 a) proper 1-cell I with I F i x = 0 or I r- F i x b) proper 1-cell I with I F i x = I F i x * = v , n n Then n v a point 2 c) 1-sphere S w i t h S F i x = 0 o r S r- F i x d) 1-sphere S with S F i x = S F i x * = v , v are 2) n n n u where v , 2 and v 2 points. Components w i t h s a d d l e points: 2 Type I component: w i s the only S Type n Fix=v Type II u w, III S, I with S , u saddle point component: veFix S on S , 1 U S u n I =Fix S 1 U S 2 U with S i n r- F i x and S=w, saddle with S S S, I. and w i s t h e o n l y component: I =w, n i n point S =0, 2 S i . 2 £ Fix , S, on S n S 1 U S . = w^ S i 2 r- F i x , Sr-]Fix= w points on S Here are S 2 L w and w, and w 2 2 are the only saddle jS. S, S, a n d S 2 a r e 1-spheres, I are 1-cells a n d w^ points. Note solid if 1 L J 1 L J a regular Klein n e i g h b o r h o o d o f any o f S, S , , bottle the manifold Proof: then If n 2 ) . Thus c a s e S 2 is a 2) does n o t o c c u r is orientable. the regular Fix N, i n case or F n N n e i g h b o r h o o d N of S i s a and cF N n are a l l annuli solid torus o r a l l Mobius 13 bands. Fix C o n s i d e r t h e components of F for a c o n t r a d i c t i o n . For then let two 1-spheres This is of case example, A be a component of that 9N and n if iF 3N in n they are 9N all annuli 9 N - F i x . A[-|Fix and A i F i x are n intersect transversally at one p o i n t . n o t p o s s i b l e i n an a n n u l u s A . Compare w i t h t h e p r o o f 3 (transversality and 4 in step 1 of the next theorem theorem). QED Corollary If they 2.6 F, are tF and F i x a r e pairwise transversal a) M is orientable b) F is a c) F is a annulus with Proof: solid In if one of transversal, the then following holds: 2-cell c a s e a) regular 9F i9F=0. n neighborhoods of 1-spheres are components are tori. In c a s e s b) excluded type almost p a i r w i s e I since and c) the 1-sphere components a r e 1-sphere S, s e p a r a t e s transversally at Type II S is or III n o n s e p a r a t i n g . In excluded a p r i o r i , while so a p r o p e r in 1-cell C cannot case case c) b) intersect one p o i n t . QED 14 A proper 1-cell I bounds and F i x be a l m o s t a disc D in a surface if I=9D-3F. Corollary 2.7 Let F, a proper C iF, 1 - c e l l or 1-sphere pairwise component of F be a s t a n d a r d c o m p o n e n t . Then any d i s c by C c o n t a i n s Proof: As only in case transversal n i F , that in F or iF and C is, let bounded standard components. b) in previous corollary. QED Equivariant Let Transversa 1ity t be Fix=fix(i) is an a m b i e n t such 3M, and that if F, an involution let F e-isotopy . are F=F 3Fix Let Lemma 1.2 to subdivision, Fix Fix a pairwise 3-manifold surface pairwise there surface transversal. transversal identity with i n M. Then to a proper 0 M. then F In the on 3M-N where N i s a ,.2 n e i g h b o r h o o d of the almost t o be t h e Proof: and are . given on on M t a k i n g F tF and F i x may be t a k e n 2.8 be a p r o p e r 0 3 F , i 3 F and F i x isotopy Theorem 0 be a p r o p e r n 3F. surface. s u b d i v i d e M so t h a t is a is disjoint i is By Proposition simplicial a subcomplex of union of 0-, the 1- and with 1.3 and respect subdivision 2-dimensional components F i x the , Fix construction certain will simplexes. e-isotopies are and F i x By . All be done taking in isotopies the star performed in n e i g h b o r h o o d s of a sufficiently fine subdivision obtained. 2 S t e p 1) By Adjust isotopies similar assume F and F i x unless F near 9F and to are Fix those . in the transversal, 9Fix are third the s t e p below isotopy nontransver s a l . In we can not m o v i n g 9F particular F Fix^ n 2 = 0. Then F n Fix consists components p r o p e r of disjoint 1-spheres and 1-cell i n M. 2 Let regular S be a 1-sphere component o f n e i g h b o r h o o d of and e a c h an annulus neighborhood no v e r t i c e s N neighborhood in 9N w h i c h is Mobius contained of n Fix . Let S w i t h N't-|F and N ' r - | F i x or on i n t ( N ) F in int(N') - S such invariant under S has a r e g u l a r i and has a a regular under is Nr-]Fix be transversal has invariant that S and F i x 9 N n band. N' a t with regular neighborhood Q no v e r t i c e s except on F i x |_j 9Q. Case torus, F N 1) 3Q n has components N, J 2 and F i x N n four and N be components o f annulus with is isotopic the identity components which 2 are are annuli. U having to a surface F' interchanged by iJ,^J . 2 no v e r t i c e s F' n N ' t. n F i solid of Let Let except by an a m b i e n t on M - N ' and s u c h t h a t a and N - F i x c o n s i s t s 9Q w i t h J^=N^ and 9A^=J^ S Then N i s two and A^ be the on 9A^. F isotopy which is x c N a n d F' n N 16 = A, A U F' N, . 2 Since i(F' N), n solid by Fix N n Case If band A w i t h these is cases Case band. two do s a d d l e F (open) J, with J, and point x-. where I and J J is a is which = A, elsewhere in Case bottle. of interchanged a Mobius arise. . and F i x N is n Let A be 9N-Fix. Only in transversally A^ with There intersect is of A , n A the are two only one = S I 2 U Proceed a saddle pairwise of H,(A)=Z and a t where y e S . Then y a Mobius one generators F(-|N is a Mobius is similar to case annulus under M o b i u s band by l i f t i n g for a arise. u n and Then N i s are 3 and 4 do not 9I=x y 2 N)=S as point in and transversally N. T h i s case represents tA n 1. represent and F i x N 4) invariant u t (F' n and S d e t e r m i n e an a n n u l u s 1-cell with n except Case components 2 N transversal. in case and S bound an a n n u l u s t(F' N) annulus. as points N n Mobius b a n d s . then J intersecting 2 n n are a solid Klein 1 using F ' N F' N, n annuli 1-spheres Case these, orientable Then N i s pairwise F' two components t h a t Proceed 3) follows n one of 9A=J. M are and 9Q has is it 2 and F i x N n J If n F N 2) torus, t. tJ,#J twice a t. (from Find n 3. self Here a curve annulus A/i) generator one t r a n s v e r s a l band and F i x N and which A=3N-Q J that a curve is intersection. is is an an bounds a J' which embedded i n A/t 17 When S i s a 1 - c e l l component of F similar t o t h e one of c a s e 9F i n N n step int(M). and 2) Adjust 1, FpiFix If veFpjFix let N subdivision proper 2-cell curve F n the self star are 1 o f s t e p 1. N i s in the i n t e r i o r annulus annulus. pairwise obtain a surface without Let (9N - F i x ) / i Take J ' loss K Fix=0. J' St(A), N the is Since representing lifts 2-cells a F is be twice a f o r one t o two 1-spheres D-and iD. D, J iD i n i n t N. P r o c e e d a s i n F only properties on s t a r sufficiently For convenience be a s u b d i v i s i o n o f M w i t h o r 2. D e f i n e 1-cell. n embedded e x c e p t s u b c o m p l e x o f K. L e t A be an m-simplex 1 F i n N. the reduced star The neighborhoods F and i F a r e n o t a l r e a d y subdividing that Take v F and a n e i g h b o r h o o d N o f F i x s u c h construction adjusts By v. in of of H,(N - F i x ) . L e t J ' transversality s i m p l e x e s o f F-N where of of N ' , i s a proper transversal F has the r e q u i r e d transversal. neighborhood neighborhood on c o n i n g t o v g i v e case of change a number o f v e r t i c e s l e t N' be a r e g u l a r intersection. Fix following of 9N i s a g e n e r a t o r and that 1 consists 1 n which We i s o t o p y may 1 the of t h i s transversal iJ, This , u s e an i s o t o p y F near F i x . i n N and F i x N in generator and be so t h a t transversal, a Fix 9M. Step By 1 above. n we pairwise may assume a l s o c simplicial assume 9F=0. and F a of F i n K with m = 0, of A i n K, t o be a l l 18 3-simplexes Stp(A), the K with A be t h e K w i t h A r- a t o g e t h e r reduced s t a r 3) F' either There p Call 1 A i n F, be a l l with t h e i r faces. a subdivision of to F such that for p(A) F'=A or A and i n t ( S t „,(A)) tr n is a Induct of on t h e no e x c e p t i o n a l Add a l l the if highest number of s i m p l e x e s the the v e r t i c e s (and Let K' K w i t h the same number of are transversality Consider the St(A)-A. subdivided already a of p:M decomposes 9 S t ' ( A ) an a m b i e n t the identity simplexes. interior isotopy of >M/i 2-simplex proper 1-simplex to with satisfy If these b is the follows St(A) are under , from i a simplexes; for vertices is F a has since D + U u fewer D. K. 1-sphere - St' (A)) D F no m=0 of and D . . + of refinement while 9St' (A) ) of barycenter exceptional F, there 1 established. two components D F of a s i m p l e x A of translates away on S t ' ( A ) . When m#2 t h i s any or t a k i n g F t o F, = (F except every m=1,2 in K'. into and T h i s determines for holds reduced s t a r s M fails where of m-simplexes it their (m+2)/(m+3) b + 1/(m+3) v of 0- theorem i s a vertex is 2-simplexes such s i m p l e x e s . is is Let p o s s i b l e d i m e n s i o n m = 0, A and v that faces. transversal, a simplex e x c e p t i o n a l c o n d i t i o n s and i s form is e-isotopic int(St„,(A)) r 2. with t h e i r projection. surface or of j- o i F t o g e t h e r Step F' o of There + which exceptional intersects the transversally. QED 19 Regular neighborhoods of F|-|tF c a n be t a k e n the in a special standard components of form. D e f i n i t i o n 2.9 Let a F, i F and F i x be a l m o s t p a i r w i s e 1-sphere the regular Then component exists with the f o l l o w i n g n F and transversally, 2) Fix n i F ' Suppose, neighborhood of S i n F and there 1) V of F and V a regular transversal iF n are annuli. Since torus. 3V i n t e r s e c t transversally, 4) If tS=S then iV=V 5) If iS#S then iV V=0 tS=S. i i s an a n n u l u s , /3, u n two p r o p e r of S these F i x - S n F intersect n V £ S and meets S and and the 1 - c e l l s or empty. above properties hold Fix S*0 then for iV. 3) c a n be arranged since i s an i n v o l u t i o n on a 1-sphere if n so e i t h e r i=id or c C a l l V a s t a n d a r d n e i g h b o r h o o d o f S. The f o u r 1-spheres has e x a c t l y (F int(M) c n Fix V Property annulus. Fix° V=0. 3) simultaneously an that properties: n n is neighborhood V V is a solid In p a r t i c u l a r in addition, iF t h e c l o s u r e o f e a c h component o f ( F i x V ) 3V. and S iF) n 3V two f i x e d p o i n t s . decompose 3V i n t o a n d (S w i t h a 2 i n four (closed) a = 0 and /3 /3 =0. 2 in 2 Call annuli a 1 f a , 2 these annuli the 20 standard annuli of V. t /3, = /3 Suppose L (a, assume 2 1 n 2 n 2 standard Relabelling, 0 ) . ca = a . Fix t tS=S. j3, ) = ( a n and 2 Fix a ? 0, n corresponding to the It 2 When follows Fix W0 2 necessary, that we n 0 , =0=Fix / 3 , n if and each neighborhood we may ta,=a, . obtain Then Fix a ? 0, n component of i 1 Fix meets b o t h a , and a • 2 D e f i n i t i o n 2.10 Let Fix S be a 1 - c e l l component o f F i x are regular pairwise transversal neighborhood V of (near S n tFix where S ) . Then t h e r e with V 9M a. n F, iF, exists a regular n e i g h b o r h o o d of 9S, c a l l e d a s t a n d a r d n e i g h b o r h o o d of S w i t h the following 1) V n F properties:. and V iF are 2 - c e l l s n e a c h two 1 - c e l l s . Necessarily Fix V n The four discs properties the is a disc, four 1-cells n V is a 2) , 4) a n d 5) a s f o r 1-sphere 3) with 9 M V F as in and 9 M V i F n standard neighborhoods. one p r o p e r 1 - c e l l o r empty. (F iF) 9V-9M s u b d i v i d e u n 3-cell. n a , , a , /3, a n d j3 w i t h a , n a = 0 , 2 n 2 2 the previous situation. standard d i s c s corresponding to V. 9V-9M i n t o /3, /3 =0 and t h e n Call 2 these discs 21 Remark of F in n 2.11 In the f o l l o w i n g i F have F and theorem, 1-sphere components S s t a n d a r d n e i g h b o r h o o d s b e c a u s e S bounds iF. F theorem c e r t a i n In is neighborhoods. the disc orientable In so the t o r u s the 1-sphere components i s and p a r t i a l again there are annulus standard theorem the c o n s t r u c t i o n w i l l made so as t o keep S i n t h i s theorem theorem discs form always. c a s e o f a n o n o r i e n t a b l e F, treated In the be annulus a Mobius band, with separately. Theorem 2.12 Let M be a 3-manifold with incompressible proper isotopy proper of M surface surface. which F is such pairwise transversal i n F. If on 3M, then the F, there iF, and no 1-spheres 3F, isotopy Then an e - i s o t o p y that i and F involution is an be t a k e n ambient on 3M t a k i n g F and Fix are 0 to a almost i n F|-| i F bound i 3 F and 3 F i x a r e p a i r w i s e may be an 0 2-cells transversal t o be t h e i d e n t i t y on 3M-N 2 where N i s a g i v e n Proof: all F bound 2 - c e l l s i F 3Fix By t h e p r e c e d i n g t r a n s v e r s a l i t y with n n e i g h b o r h o o d of contain innermost n^F. theorem there t h e above p r o p e r t i e s e x c e p t p o s s i b l y no in i n F. By Corollary 2.7 1-spheres those s a d d l e c o m p o n e n t s . L e t S be a 1-sphere i F , that is, there is a 2-cell D i s an F f- in 2-cells of /.Fix F n i F with 22 D F=3D=S. Since n F. If F is compressible, iS=S t h e n we may assume Let V be a neighborhood e x i s t s be the There and w i t h Dx1 F F n ' n a F ' C (F iF) - n F' identity Such D is innermost Then collar it Lar~]Ci=0. follows (Dx1) i(Dx1)=0 n u Since a i n F and i F . L e t a F'=(F-(B Sx[-1,1])) Dx1. S and F', M is and F a r e a m b i e n t on 3M. By S. = Sx[-1,1] F n thin Consider transversal. 2-sphere, 2- c e l l s Since sufficiently pairwise the of o f D=DxO w i t h = DX[-1,1] a * 0. n i(Dxl)=0. t S bounds a d i s c Dx[-1,l] 3DX[-1,1] for neighborhood s t a n d a r d a n n u l u s m e e t i n g D b u t n o t B. is a bicollar B in iB=D. standard since S bounds a 2 - c e l l and F i x a r e almost irreducible isotopic induction, all and Then u iF' that and D u B by an i s o t o p y 1-spheres is a being bounding c a n be r e m o v e d . QED Def i n i t i o n 2 . 1 3 A 2-cell and B in a 3-manifold 3B d o e s n o t bound a 2 - c e l l is essential if in an 3- m a n i f o l d a n o n s e p a r a t i n g p r o p e r The f o l l o w i n g theorem a l s o 3M. In 2-cell is appears in it i s proper irreducible essential. [3], 23 Disc Theorem 2.14 Let M be an i r r e d u c i b l e Suppose M h a s an e s s e n t i a l 2-cell essential 2-cell and e i t h e r B t B = 0 o r tB=B. in the point 3B=3B is case a n d B and B the i d e n t i t y Proof: By B and B 2-cell ambient 0 of B. I f 2.12 isotopic of proper particular then 3 B show to Let D n iB=3D r i 0 n obtain D and B w i t h B, consists how be an and iB *0). o By outermost t h e component o f 1B-1D t h a t 0 be and and isotopic by an i s o t o p y that D'=tB-tD. 2.6 If there B n iB is only. Assume tl#l empty of suffices D define does c (in n it B: or B iB*0 B^ w i t h fewer disc an transversal, either induction is to 1-cells in c B with 3B. D' t o be t h e c l o s u r e not contain tl. See L e t V be a s t a n d a r d n e i g h b o r h o o d o f I a n d l e t a , , j3 D'*0. n o 1 - c e l l o f B a n d 3D-I of 2 and n t h e n one c a n t a k e t3B =0 a new 2 - c e l l iI=1 d e f i n e a o n B Fix=0 iB and F i x p a i r w i s e If 3. an 1 - c e l l o f B o r one Corollary 1-cells tB = I a p r o p e r Figure 3B is on 3M. Theorem essential there i. B and F i x a r e t r a n s v e r s a l i s a proper n a r e ambient 0 Then 0 In t h e f o r m e r c a s e B Fix in the i n t e r i o r 0 B . B r- M s u c h t h a t n latter 3-manifold with i n v o l u t i o n s t a n d a r d d i s c s of V a,|-|a =0, with 2 Consider B ^ D M / S M D ' ) - int(V) a n d B =D 2 u ( iB-D' ) . a /3 D^0 i n n F i g u r e 3. Then B by i n iB, (B iB) c induction essential or then irreducible I, point thin 2 and note bicollar n B ' = (Dx1 2 isotopic u I 2 n 9M. S i n c e E bounds a 3 - c e l l . M isotopy taking essential. c Fix I If which n B 2 I= 11 we n a and is tB ' c B n Dx1 meets essential iB - a sufficiently n tB have is necessarily l i l = 0 consider 2 is to iB. o f D=DxO s u c h t h a t D x [ - 1 , - l ] I in and B ' of B, i s n o t Using U 2 E If 1 1 done iB would n o t be e s s e n t i a l . iB)-(Ix[-1,0) D') to B B j-|iB =0. are 3-cell U 2 If we This 1 is 2 that F i x B Dx[-1,-1] of case an ambient o f I o r a l l of I. bicollar B otherwise So we may assume B 2 at 9B, bounds a 2 - c e l l 3-cell construct iB =B i f B, i s essential arrive t h e 2-sphere d o e s n o t meet the - i. n a,. since is a Then it is I. QED 25 II. INVOLUTIONS #3. Some Involutions -The classification be u s e f u l disc ON THE 3~CELL AND THE SOLID TORUS in the theorem will classifying the Definition 3.1 be C be t h e used on a on a be t h e the next reduce the solid standard :|Z|<1} be t h e standard 2-cell standard D ={zeD :z=x+y•i, Re={zeD :z=z} £ D , 2 two 2 involutions K(Z)=Z 1 rotation by which is free. above which is Then a . K = f c . a = - K 1-cell s p a c e s as follows. reversing with orientation is fixed preserving conjugate to K by a 90°. 2 set 1-cell orientation On D : J c ( z ) = z o r i e n t a t i o n Re. 2-cell a proper on t h e p o i n t s ± 1 . a(z)=-z and f i x e d p o i n t a 2 I m = { z e D : z = -z} Define torus j- D , a p r o p e r 2 On S : y^O}, 2 + of 1-cell D ={zeC be t h e one Let 1-sphere 1 to of 3-cell. standard 1 The problem torus be t h e 2 will chapter. :|z|=l} T =S xS fixed to on a s o l i d t o r u s {zeC 1 = 2 1-cell in complex n u m b e r s . 1 set theorems involutions I=I =[-1,1] S of involutions involutions classifying Let proof of a(z)=-z one p o i n t which 0. reversing is Then -k with orientation is conjugate fixed set one preserving with to ic and has 26 fixed s e t Im. On I : r ( t ) = - t one p o i n t t h e map p : S x S 1 p:D xS 2 orS^S >D xS 1 >S xS 1 1 1-sphere is orientation reversing with fixed set 0. Define map which 2 1 1 >S xS 1 1 by 1 similarly. p(z,w)=(zw,w) Define by co (z , w ) = (w , z ) , w h i c h h a s and involution fixed set one {(z,z):z}. Lemma 3.2 There are annulus S xl. 1 five involutions They are: 1) up axid to conjugacy which is on an orientation preserving and f i x e d p o i n t free, 2) a x r w h i c h is orientation reversing and f i x e d p o i n t free, 3) K X T w h i c h is orientation preserving with f i x e d set two orientation which reversing is orientation points, 4) idxr w i t h f i x e d s e t a 1-sphere, reversing with fixed set which and 5) two is Kxid proper 1-cells. Proof: set When t h e d i m e n s i o n o f t h e f i x e d s e t i s one t h e f i x e d separates. characteristic Proposition In the argument other given case in part use 4) the of Euler proof of 1.3. QED 27 Definition For (see 3.3 the Figure 3-cell D xl 2 having 2 having J,=KXT jo=axr having 2 and jo the following involutions 4): j =idxr j define are preserving, f i x e d set fixed set j , i's one p o i n t reversing conjugate 2-cell an u n k o t t e d f i x e d set orientation a proper to D x0. 2 1-cell RexO. 0x0. while is orientation j , ' = a x i d w h i c h has fixed set 0x1 . Theorem 3.4 An i n v o l u t i o n All involutions Proof: Let on a 3 - c e l l on a 3 - c e l l t be an and P r o p o s i t i o n is are involution conjugation on 3 - c e l l separates. J2 Let E. to j 2 , ji or j 0 . extendable. A p p l y Lemma 1.2 1.3. 2 Suppose conjugate Fix E, 2 #0. and E Since 2 Fix be t h e proper components, j i Figure 2 is and ^ , £ = 1 , E=E j o 4. 1 U E 2 Fix with 28 Fix =E i n E . If 2 Fix were compressible then let B . compressing disc in E say. 1 f Using a Mayer-Vietoris Then s e q u e n c e we see [3B] H,(E,)©Hi(E ) = 2 H,(Fix a 2 . iB c o m p r e s s e s F i x in must be 2 in be E . 2 trivial 2 ). Hence Fix is a proper 2-cell. We show Construct closure an given. a point this conjugate BE^Fix to extendable Extend h to o b t a i n the over 0 an from 0 case to referred Fix to in part i | 3E shows F i x 3 E n 4) has Let disc of f i x e d set 0 or which this latter is case orientation Fix * 1 has one plane and j Fix 0 | . In the is S xl. |_jFix =0. only, By the call conjugation it v. u 3E/t h :3E 0 extendable h to star(v) >D xI-star(0,0). [9] t has no f i x e d p o i n t since 2 former =0 and i n 2 Fix a Lefshetz a conjugation Extend 2 applied In the number argument is a 2 case h is 0 a i projective >3(D xI) s u b d i v i d i n g we may assume s t a r (v) r-| 3E=0 and E = 1.3 2 fixed point so t h e r e argument 0. 1 Suppose to conjugation. Proposition so F i x the Extend 2 characteristic of In >D X[0,1], a the isomorphism 2 fixed points. reversing is and t h e n c o n e 1 i case 2 1 2 Fix t o get proof D =D X[0,1]. 3D, - D x 0 . h:E, Euler of . we may assume the The 2 the isomorphism =0. j closure i s o m o r p h i s m by e q u i v a r i a n c e Suppose to is isomorphism h of conjugation is i between c| given. By int(star(v)) conjugation E T h i s c a n be done by Theorem 1 i n on 3 E . F i n a l l y cone to v. 29 Suppose Consider the involution [16], Let in u We with F i x of E. induced has with by It t. is 1 there £ 3B r- F i x 3 E a 3B 3E=C. is u 1-cell C this with fixed point free V is a n n u l u s and t|(V N) Using an these construct facts a disc Construct the a conjugation to j , B cB In >DxO. h 1 ( (3D D n + )x0) . and + the and that Then B tB u then in can be This u B embedded i n E Moreover, C if N is E-N orientation is also if is a star a solid preserving. fixed point it free. is possible to follows. Construct an equivariance to claim. as extend conjugation c o m p o n e n t s . We e x t e n d t o an components and n >D xO 1-sphere 3E Fix*. c c l o s u r e V of in B u the required isomorphism punctured these disc and D i s c Theorem 2.14 as a is tC=3C=3Fix t h e n we may assume n torus N 3B' a B 1-cell. c l a i m note then is n n of n B 3E B iB=Fix. neighborhood i| Fix is n (nonproper) such that with To e s t a b l i s h n a B E E with c and s u c h t h a t arguments one u n k n o t t e d p r o p e r claim is 3E. By standard with 1-sphere. B. B' 3-sphere one u n k n o t t e d f i x e d set to points. Waldhausen of 1-cell a two of one component of a is By a r e s u l t f i x e d set 3B t h e 3E n is but giving shows F i x above F i x all interior 3E i ' the p o s i t i o n with respect and removed E U be a 2 - c e l l 3-cell By involution general the #0. double E E ' i this B Fix by extendable separates E isomorphism over by e q u i v a r i a n c e to the case we s e t C = into two 3-cell one of these other, giving a 30 conjugation E >DxI. QED Lemma 3.5 Let F be a 2 s i d e d s u r f a c e be an involution sides of F' n Then F iF=F and s u c h t h a t is ambient isotopic an t invariant i t interchanges to a s u r f a c e F' with iF'=0. Proof: by F. on M w i t h i n a 3 - m a n i f o l d M and l e t Construct using a star n e i g h b o r h o o d of bicollar F. FX[-1,1] Then c o n s i d e r of F=FxO F'=Fx1. QED Remark 3.6 Suppose tF=F for 3-manifold obtained FX[-1,1] M £ by by c u t t i n g distinct F x[0,1]=Fx[0,1]. N 2 d:F,=F,x0 a 2-sided has Since involution k on N w i t h i for does an iF=F. not t is M a l o n g F. That canonical t = k / d . Note k then the N be is, i n d u c e d from M. identification. there k.d=d.k bicollar. k i n d u c e s an is a and Let Then canonical and k ( F , ) = F Conversely involution the replace F,x[-1,0]=Fx[-1,0] simplicial interchange involution Let a subdivision 2 M = N/d. F. copies > F = F x O be t h e 2 surface if i 1 iff k.d=d.k in M with 31 ^4. Involutions Definition Let on t h e Solid 4.1 V be t h e solid torus V = D x S = { ( z , w ) : | z | < 1 , | w | = 1, 2 Torus z,weC}. 1 Define the following J =/cxid having involutions fixed A having J =p.(Kxid) M set the f i x e d set {(s-e J2 =idx/c having D Recall fixed J p=jo. ( i d x K ) ' h a v i n g D Definition on V (see annulus ,e 2 7 r i t Figure 1 band ):0<s<1,-1<t<l] set two 2-cells D x±1 . fixed set a 2-cell 2 and a point D x1 2 j =axid having fixed set one 1-sphere J2£=KXK having fixed set two 1-cells Rex±1. J2p=ax/c having fixed set two p o i n t s s J =/cxa fixed point N JQ=idxa So fixed point J (z,w)=(zw,w) J the fixed point set involutions the 1.3, V since 2-dimensional with None is different the orientable, fixed sets fixed above fixed sets are sets involutions or 0x± 1 . reversing. and o r i e n t a t i o n or for the type. fixed Recall orientation are orientation by subscript point 0- reversing. conjugate free Proposition with orientation type. preserving. The involutions are Ox-1. 1 (z,w)=(zw,w). orientability 1-dimensional of D P u OxS . and o r i e n t a t i o n free and M describes free 5): RexS . the Mobius 7 r i t 3.1. or Those preserving. since Using all have V=D xI/d 2 32 33 where d = a x ( r | 3 I ) , d e f i n e d as involutions f i x e d set M Theorem D p '=idxT/d having i and i' are isomorphic fixed point free of and conjugate. the Proof: 2 sets Let or can p be Rexl/d u listed i be an on V i s involution is B=D x-1. 2 2 i and an i' type, w i t h nonempty 1 are fixed then t and conjugate to one on V . any We show given isomorphism h it conjugate may assume i involution h 1 is essential i' are of the .i.h for conjugate 2-cell of V w h i c h So by a p p l y i n g t h e D i s c Theorem 2 . 1 4 , the point above. of there if on V = D x S same o r i e n t a t i o n i n v o l u t i o n . For by D f i x e d set D x 0 0 x l / d . to a standard i and J M o b i u s band involutions An i n v o l u t i o n involutions V j 4.2 If nine to follows: J '=/cxid/d having J conjugate takes and B B to 0 replacing a suitable h, we satisfies: Case 1) iB=B and B i n t e r s e c t s Fix transversally at Rex-1 Case iB=B and B i n t e r s e c t s Fix transversally at Ox-1 2) or 3 ' ) There is an 0 iB B = 0 n isomorphism ( D x l ) / d >D xS 2 2 1 where d=idx(r|gj) 1 7T t is to given B. Write isomorphism by d ( z , t ) = ( z ; e a l s o B=D x-1. 2 h we may ). In assume The isomorphism takes case that 3') by D x-l/d 2 adjusting the iB=D x1=(D xO)/d. Call 2 2 C.=D x[-1,0] and C = D x [ 0 , l ] . 2 The c a s e 2 + 3') s p l i t s into two cases: Case 3 ) i B B = 0 and iC =C. Case 4 ) t B B = 0 and tC =C . n + n We show f i r s t , + + i f the i n v o l u t i o n t falls case 1) i t h a s f i x e d s e t t h a t of J case 2) i t h a s f i x e d s e t t h a t of j g , case 3) i t i s f i x e d p o i n t case 4) i t h a s f i x e d s e t t h a t and we show s e c o n d , 4) then case the nine sets N and j of J r j ' 2 fall are, ^DP' ^2C o ^2P' r i n t o t h e same c a s e This will involutions that can a r i s e Q complete cover a n d none o c c u r s 1) - the proof all possible i n more t h a n one c o n s t r u c t i o n s done f o r i a r e t o be p e r f o r m e d f o r i ' a l s o even i f not e x p l i c i t l y stated. Use a p r i m e ' to denote corresponding construct. In Case 1): involution restricted since the standard as J , M 1) - 4 ) . All the i and t ' t a n d i* a r e c o n j u g a t e . because fixed if free or J a into: X The on Fix=fix(i) standard Fix(X) disjoint is D xl Fix(X) 2-cell.. 1-cells t the is j 2 Fix of is on D xl/d induces 2 property proper i s transversal involution a with 2 to D x3I. 2 involution and X.d=d.X an when 2-dimensional t o B. So X i s c o n j u g a t e t o the 3-cell. In particular o b t a i n e d by i d e n t i f y i n g two i n t h e b o u n d a r y o f t h e two c e l l so Fix is 35 an annulus or Suppose there is extend h so a a J in Mobius determined X|B conjugation into be an that h:B (open) same band if 3Fix(X). D x9I Then which Fix Therefore, h 2 l and are may 3Fix(X) interchanged (3D x-1) is an - 2 by so d'we x3I. of determined id(J) conjugate d >D 2 2-cells and are Using component 2-cell J X|B' >B'. two h(J)=J'. the by and conjugation 2 selected band. given. 3D xI Let are is to X. d(J) i' a decomposes under Mobius annulus 3Fix(X) Fix(X) if and are in the can be extended J and Fix same is 2-cell to a conjugation h:D x3I 2 Extend D xl 2 h over into. This a l l of extend a conjugation F i x ( X ) one Then gives u D xl between t and In case u that other cell h defined property hence 'x3I Fix(X'). 2 2-cells the conjugation and 2 the to extendable of >D for induces As 2): in case the X Fix(X) is proper 1-dimensional involution Suppose Since and a X|3B t' is j , on 1) involution is decomposes equivariance. 3(D xI). By 2 3-cell, h the extends conjugation to on D xl/d on D xl/d 2 t'. an there by on the a induces standard 3Fix(X) of 2 the orientation with so 3-cell. given. conjugation is D xl X|B h:D x3l 2 involution So >D preserving, X.d=d.X X and i is Fix X|B' 2 , x3I h on conjugate is are a D x3I. 2 to the 1-sphere. conjugate with extends 2 so h.d=d'.h. to 3(D xI). 2 36 By the conjugation extends to all extendable of D x l 2 property a n d hence for 3-cells, h i n d u c e s a c o n j u g a t i o n on D xl/d. 2 In Let case i and i ' 3): be isomorphism of >B' >B' c'B' type the is same, h:C D x S =C |_j C . by 1 each has by + 4): t|C set a (B and t|C a and conjugation the to 2-cell type have + h:B iB iB) Fix=0 conjugation the orientation 9C + and t h e n t o an extend to i|C property h:E have fixed sets. 3). Suppose 9C >3C '. + + so t that Construct a In of 3 - c e l l s , two view it of suffices Let G=3C + Fix(t|C ). + i s conjugate two components one o f w h i c h , isomorphism a n d t | C . must Arrange n o t a t i o n as i n c a s e u + + follows o r two 1 - c e l l s . isomorphic extendable i|C 1 - c e l l or a p r o p e r so F i x i s one o f : two 2 - c e l l s , show h e x t e n d s t o a c o n j u g a t i o n Case 4 . 1 ) on 3 - c e l l s so it n >B' t'B' u and l e t F i x now d e n o t e an u union a p o i n t , t'|C " conjugation into an a Finally a proper c' have i s o m o r p h i c f i x e d s e t s . + Since coning. Moreover u same o r i e n t a t i o n points, to t o a l l of point, Fix=Fix( i |C ) F i x ( t | C . ) . the Construct and t | C . a r e i n v o l u t i o n s + Since + extend h extends >C' + fixed 2-cell. and type. free. equivariance. + In c a s e orientation by e q u i v a r i a n c e . u isomorphism 2 same h:B h:B iB u The i n v o l u t i o n must be f i x e d p o i n t >E' and to E, by j 2 . Fix contains decomposes B. G Extend h to equivariance to a 37 conjugation h:G Case 4.2) such that >G'. i|C + i s conjugate Fix=Fix/i misses B/i. 1-cell J , in J iJ=Fix. In a d d i t i o n , containing B but not i B . D e f i n e n and extend selected for J u >i'J' 0 tJ B U h 0 and Case over double a J 1-cell isomorphism t J . so i f may t h e (open) of is is + a Notice select h':G G/t the are fixed t o be e i t h e r 0 reversing. J u to and u Hence ij) t o an to G. j h isomorphism 0 . Then induces isomorphism extends and >J' 0 we c o u l d have h a n n u l u s E-(B band with h :J orientations conjugate Mobius G a component E o f 3G orientation >3E'. This covers isomorphism or i|C iB))/i i s o m o r p h i s m 3E we 2-sphere an a p p r o p r i a t e an u a So l i f t i n g determines to J is > E ' . E x t e n d by e q u i v a r i a n c e 4.3) u t J instead, preserving isomorphism h ^ E G u J'|jt'J', extends E=(G-int(B J gives equivariance h :J orientation h| (G/t)-(B/t) by t o j , . G/i an t o a l l o f E. lifts to >G' e x t e n d i n g h . By c o n s t r u c t i o n h ' an is a conjugation. QED Let projective to C^P the space space t o a p o i n t (B ,0)/j . 3 be o obtained v ^ . Then by coning (C^P,v^) The d e s c r i p t i o n s o f t h e s t a n d a r d on a s o l i d t o r u s V c a n be u s e d t o compute V / j . denote F i x / j and r e c a l l Corollary 1.4 i n t h i s is a real isomorphic involutions Use Fix connection. j to 38 V/J is a a solid torus D xS 2 with 1 Fix an annulus Re(3D )xS . 2 1 is V/J ' M a solid Klein bottle D 2 + XI/-KX(T|31) with Fix the M o b i u s band R e x l . V/J 2 V/J D V/J s is D is ' P is V/J 2 C V/J 2 P Fix=v i u a 3 - c e l l with Fix the D xS 2 point v, a 3 - c e l l with Fix is a b o u n d a r y c o n n e c t e d sum v and a with F i x 1 is the two p r o p e r of 2-cell. 1-sphere OxS . 1 unknotted C,P 1-cells. and C P 2 with 2 V/J S =D S, + n with F i x a s o l i d torus (i.e.) 4 C,P two 2 - c e l l s . 2 P =D xS 2 + / ((z,1)~(-z,1) v,=(0,1), v =(0,-l) 2 , ( z , - 1 ) ~ ( - z , - 1)) and D x i 2 is the where c o n n e c t e d sum disc. V/J V/j N Q is a solid Klein is a solid Corollary If V/t of D XI/KX(r|31). 2 torus. 4.3 is i isomorphic type bottle to the an involution one of the f i x e d set up t o i s o m o r p h i s m . on a s o l i d t o r u s V t h e n V / t spaces V / j above. and o r i e n t a b i l i t y The type of is isomorphism i determine 39 Example 4.4 j Q i s not c o n j u g a t i o n extendable because determines a c o n j u g a t i o n s:3V s(z,w)=(z,zw) >3V f o r j | 3 V that does 0 not extend t o V. C o r o l l a r y 4.5 The o r i e n t a t i o n r e v e r s i n g extendable. If V and orientation preserving V involutions are solid t involutions are c o n j u g a t i o n tori with conjugate i' and respectively then the i n v o l u t i o n s a r e c o n j u g a t i o n extendable with respect to the c l a s s of isomorphisms 1 ) extend to conjugate t o j Q 3V isomorphisms > 3 V that: V >V , u i conjugate to J 2 >V' Fix', extends involutions j only. extendable In f a c t to an isomorphism H':V V-Fix an >(V-Fix)/j isomorphism. lifts to 1 ) = j , that h' >V with H'.j.(H' 1 ) = j . Now >3V/j. We show, f o r be induced by i n c l u s i o n . p | ( V - F i x ) i s is f o r the i t s u f f i c e s to show >3V an isomorphism with h ' . j . ( h ' h extends to an isomorphism H:V/j p|Fix t c h' induces an isomorphism h:3V/j j, case f o r the case u P r o o f : I t s u f f i c e s to show c o n j u g a t i o n given h ' : 3 V the or j<,. 2) extend to isomorphisms V F i x standard for >V/j. L e t p:V a Check H':V-Fix double that >V-Fix each >V/j cover and H.(p|): and thus obtain a conjugation. Let W=V/j. a proper When F i x submanifold The e x t e n s i o n s J D and P the J For . M boundary of ( 3 V ) / j of H of J of also order h(9D) For 2, order by c o n i n g o v e r J p ' 9D xi is 2 2 2-sided in h:9W Fix a 9W. (0,2)eH,(9W). component of 2-cell u D in W that i- w s proper J Klein a disc has -'s' 2 bottle. D -'o' For J represents over that the separates 1-spheres is above. given. disc j A 1-sphere and the For 2 is element c , let 9W-Fix that two components e a c h containing one the bounds a property misses F i x . Use that this it in D'. W-(3WyD). 9W represent , N the 2 as into on i n H , ( 9 W ) = Z © Z . S i n c e h(9D) 1-sphere be defined 3 D' , A 3-cell Such Fix ^ for D and t h e n u 9W Fix only proper Proceed decomposes clear , two h is bounds an e s s e n t i a l >9W' Fix' u P an e s s e n t i a l element Extend 2 that 9(V/j). h are and J n unique of ±0 n o t i c e 2-cell proper to extend h. QED Remark 4.6 Consider the d=/cx( T | 91) . Then J -/cxid/d w i t h fixed set a point 1-sphere solid a and a are a Mobius and a 1-cell, the bottle = /cxid/d w i t h f i x e d f i x e d set a 2-cell Klein 1-cell, and j g only = five band, j c p V=D xI/d set J D ( an a n n u l u s , , = = -Kxr/d axid/d where 2 with involutions J idxr/d on a s o l i d = with with f i x e d fixed M set set a Klein bottle, one up to conjugacy. given for the solid however, only combinations occur. torus. i s very Since similar d is case 3) does not a r i s e and reversing, the The proof to the orientation in case 4) that were d i s a l l o w e d p r e v i o u s l y can 42 III. #5. EQUIVARIANT ANNULUS Annulus 5.1 proper annulus decomposes M i n t o such torus V=D xS M = 3M 2 trivial, if A and a s u b m a n i f o l d M 1 0 that: there exists B A=3B A n trivializes Note different essential surgeries if boundary o n v 9 0 1-cell = V n A proper in 2-cell B V with c A. A nontrivial. Call V a solid A separate does not M or components of M t h e n A i s torus that if 3A i s in nontrivial. 5.2 a nontrivial annulus. Mobius Call incompressible an F be a s u r f a c e performed Fix are neighborhoods (see proper incompressible annulus proper Mobius an band band. later F'=Fx1 where F X [ - 1 , 1 ] and V, a nonseparating call that an e s s e n t i a l Let U A. Definition Call 0 a nonseparating n Otherwise iF' A in a 3-manifold M i s a solid M = M and THEOREMS Theorems Definition A AND TORUS is and S a component we will wish a bicollar transversal Definitions at of of to F n i F . replace F . To i n s u r e least 2 . 9 - . 10), the In in some F by that F * , . standard following lemma 43 is used. Lemma 5.3 Let A A V be a s o l i d t o r u s and and A , 0 0 c n i A S . = s be i ' annuli iS =S , 0 0 Then t h e r e 0 and Fix is intersect in (A 0 u 3V A, ) n t (A 3A = S | _ | i ' 0 transversally = (A n V such that c n 0 A and F i x 0 with A i A = S int(iA ) 3 1 S, s 0 L J A , )=S 0 involution. and t h e a 0 u u A, 1-sphere other Let S 2 r A,) IA and component S . 2 A similar are with >V an a proper annulus A 3A h a v i n g one component i n is i:V 2-cells Proof: small By statement and are holds if 1-cells. transversality regular assume one of neighborhood the V is a 3-cell See F i g u r e of Fix, N of and 0 u A , 0 A, 6. by t a k i n g a A A , u cA 0 sufficiently u tA, we may following holds: 1) F i x N = 0 n 2) Fix N n consists of two disjoint 1-cells I. with 44 exactly one p o i n t 31^ i n S or 3) Fix N is S and t h e other in Further, there is n 0 such that is of U i A S. 0 B . 1)~3) be There A" iA"=S. n a regular 0 is an S i n Let A' is be t h e the is annulus an a n n u l u s A " U int(V) 3N n one b o u n d a r y n e i g h b o r h o o d N' an a n n u l u s , hold with respect is Then A = A ' A " in component n n the Then t h e r e other int(V) 9N. 0 Let 0 such t h a t iN ' =N' , A ' = N ' A and p r o p e r t i e s A an a n n u l u s and t h e 0 to is int(B) with component of 0 1 S 0 iA '=N' u N c 0 3V n N'. which in V-N' A of the closure of N' - tS=S and F i x B n such t h a t B-S 3A" = S that 2 U S meets c and iA . 0 desired-annulus. QED Remark 5.4 A solid annulus. the U is U is essential a s o l i d torus a solid Klein separate Moreover, that is is a twisted but it I-bundle over does not an separate 5.5 If that The a n n u l u s bottle boundary. Lemma If Klein t h e n U has no bottle then one annuli. U has no e s s e n t i a l annuli 3U. suppose A' is an a n n u l u s a n o n s e p a r a t i n g p r o p e r d i s c D of exactly essential nonseparating 1-cell contained U of in 3U s u c h intersects A' in A'. is an If A 45 incompressible proper annulus disjoint solid A may be t a k e n torus which trivializes from A' then t o be the disjoint from A ' . Proof: Suppose A i s an e s s e n t i a l proper as in nonseparating 2-cell the statement.) incompressible a n n u l u s . Then l e t of U. (When A ' is given, Make A and D t r a n s v e r s a l . adjust D so that A D n D be take D Since consists of any A is 1-cells only. If A (-)D=0 then A i s removing a s u f f i c i e n t l y U. This contradicts If A D*0 n disjoint from let B be A' if A D'. n I that B, meets possibly, A. Let int(B). in n e i g h b o r h o o d of I outermost is I obtain Assume now t h a t separates regular an A' If n moving small given): n must a d i s c D' does not decomposes the c a s e where U i s meet D from n 3U the a n n u l u s . It D (and is a n in A, then w i t h fewer bound a 2 - c e l l of of so B A = 3 B A = I bounds a 2 - c e l l V be t h e c l o s u r e 3A 2-cell by an 1-cells in A. into two a solid Klein follows in the that in Then t h e component of an a n n u l u s and two M o b i u s b a n d s . However, 3B 3U o b t a i n e d by incompressibility. 1 - c e l l and B 3 U = 3 B - I . isotopy contained in a 3-cell U-A annuli or bottle, into latter case 3V 3U a n n u l u s and V i s a s o l i d t o r u s w i t h t h e p r o p e r t i e s is an making A n trivial. QED 46 We n e x t annulus spirit to the proof 1) M be an i r r e d u c i b l e there 3ALJ I 3 A = 3 A or 2) 0 1 is J i 3A there iA,=A annulus of the torus be an e s s e n t i a l 0 partial theorem and t h e in theorem. A n n u l u s Theorem 5.6 Let A the t h e o r e m . The p r o o f s a r e o m i t t e d . They a r e s i m i l a r Partial Let state 2 transversal 2 a n n u l u s w i t h 3 A r-| 13A =0. T h e n : O 0 essential annulus A w i t h A i A = 0 and n 0 are iA =A , l f an 3-manifold with i n v o l u t i o n i . two d i s j o i n t and essential 3(A, A )=3A u 2 0 u i3A annuli 0 A,, A and 2 Fix with is t o A , and A . 2 Example 5.7 The i n v o l u t i o n RP xI=D xI/d 2 2 3D xI=S xI. 2 1 2 where d=axid Fix=(Re {i})xi. u band satisfies There i s an a n n u l u s , transversal Jcxid on D x I A iA=0 n to F i x . i n d u c e s an i n v o l u t i o n i s an i d e n t i f i c a t i o n No e s s e n t i a l t on d e f i n e d on annulus or Mobius o r iA=A and A a n d F i x t r a n s v e r s a l . however, with iA=A but it is not 47 Example 5.8 Consider over a the nonorientable torus, where twisted I-bundle Ixlxl/d d = ( T | 9 1 ) x i d x r i d x ( T | 3 1 ) X T . The u involution t=idxTxr/d h a s M o b i u s bands b u t no a n n u l i Ar-|iA=0 o r iA=A and A a n d F i x A with transversal. A n n u l u s Theorem 5.9 Let M be an i r r e d u c i b l e Suppose A 3A =S So2 0 0 1 u 0 i s an e s s e n t i a l where, respectively respectively R,*R 2 R , assume a l s o with either that the R, M is orientable of S or c R 0 1 =S 2« 0 annulus or 1 A may be t a k e n R 1 f 1 f if space. Mobius band A transversal R |_,iR |_|CR . 2 i. r_ R i n c o m p r e s s i b l e , and iA=A and A a n d F i x l u Let 0 3M w i t h S i i s not a p r o j e c t i v e n u is involution o r M o b i u s band i n M w i t h component A iA=0, 9A t9A with i s a Mobius band, i s an e s s e n t i a l in both cases If 0 annulus c ^2- Assume R, 0 2 Then t h e r e and be 2 S if A 3-manifold 2 t o be an a n n u l u s . 48 #6. Equivariant Lemma Torus Theorem 6.1 Let M be incompressible an irreducible torus. Let F be a interior of M and W a r e g u l a r a torus. Then 9W i s Proof: If W an is not ^(M). twisted inclusion the torus. to that is with R is of I-bundle p(TxO)=F. incompressible >M, t h e s u b g r o u p . Then p a lens over U in M determines Consider p:H corresponding torus in Necessarily incompressible U s u b g r o u p of 2-sided n e i g h b o r h o o d of bottle then M=W U w i t h U a s o l i d t o r u s . an orientable tori. 1-sided K l e i n an i n M w i t h 3W The solid containing F orientable. a 3-manifold space. p 1 But 1 T and M. i s an i n d e x 2-sheeted two covering ( W ) = T x [ - 1 , 1 ] where T (U)=V 1 U V is 2 two disjoint M and hence H c o n t a i n s a torus. QED Equivariant Let T o r u s Theorem M be an Suppose M c o n t a i n s following (I) bottle 6.2 irreducible 3-manifold an i n c o m p r e s s i b l e with torus. involution Then one of i. the holds: There T in is int(M) a 2-sided incompressible transversal to F i x torus with T i T = 0 n or or Klein iT=T. 49 (II) and M=V. , V , u tV =V i i and i n A . , annuli = A and V - n U . ^ A j , Figure bottle i A is 1 a M=V when restricted = 9A i bottle V 2 = (T (IV) 1 U V 2 U tT) n Proof: Theorem such that t is of and V . n Let T 0 2.12 V,, i T be V = S - int(V) 2 2 = tori U, U., n au.-Aj and tA_. . u See S. See incompressible and t h a t 0 no tT 0 1-spheres Fix V is a torus 0 in are n a torus tW=W, iT=T and T e x c e p t contained and F i x in T 7r,(V). and I-bundle over 3W 1 of 7. n Fix 1-sided iT 3 W = 3 V = W V and component S o f T , a n a twisted to assume is T, tori preserving and F i x S * 0 . Figure with solid a generator 2 transversal an There Klein preserving. are two a n n u l i . n or orientation £ int(V) are torus V and F i x 3 V = 0 where W i s is 1-sphere transversal 0 solid orientation i a s o l i d torus Fix is V transversal U c jV^ where V , , t o any M=W V tV=V. T . V., incompressible V under £ W and V i s possible in n av^A-LjiA., to F i x . T with T pairwise and are with = 3iA. = V , i i f 2-sided s t a n d a r d n e i g h b o r h o o d of T i =±1, n invariant are n A^, V. U_. = iA each n i transversal (III) V , and u\ 7. A,|_|A. Klein where tU.,=U,. There are A U . ! LJU, U iT 0 in for a T. int(M). By almost pairwise bound 2-cells (II) (III) -1 Figure As a first step components a r i s e . possible. In transversally in T t T 0 n and S 2 o n iT is 0 and are Fix regular once. transversal. n e i g h b o r h o o d of S, S saddle components and are intersect can be o n l y one neighborhood of it is Suppose T 0 n i T 0 a torus T' be r e g u l a r invariant n e i g h b o r h o o d of is since 0 component i n 2 Then t,T=T and T Let under . and b o t h since a bottle. a 2-sided torus S 2 1-sided N. satisfying regular N be a If 9N is (I). If First we (IV). a Type II component S S , . t o T=T u 0 n e i g h b o r h o o d s of with i T ' = T ' . Let S and S, under i s u c h t h a t N = N (S) LJ N ( S , ) S S, and s u c h t h a t T N ( S ) U U S =0 is at 1 u invariant isotopic S S U T = A i A . Then Klein T c o m p r e s s i b l e we a r r i v e construct component S Let T is a solid i n c o m p r e s s i b l e then both there bound an a n n u l u s A i n T S transversally and N ( S , ) since a Type III intersect 3N i s c a s e s where and Type II cases one p o i n t , the o• Suppose T S, handle O n l y Type III both at we 7. N N(S) respectively, is a regular and T N ( S , ) n are 51 annuli, N(S) i F n is x 2-cell and N ( S , ) F i x NfS,) are n disjoint that Klein Fix 1 i s a proper bottles. open 2 - c e l l meet Mobius band, N ( S a ) i F n 1-cell. there are is x N(S) there and K of two disjoint 2 proper a Both By t r a n s v e r s a l i t y components K, and 1 1 and are two N(S)-(T iT) u open 2-cell 2 components L, By and L considering A=(K, K u 2 C=3N T. the Fix In u L 2 ) n effect 3N is irreducibility contain Since L, the The c l o s u r e of A n by t h e u of N ( S , ) - ( T 2 particular t|E to is invariant and c o n t a i n s F i x Fix 1 n a 3-cell E is taking N with D w i t h 3D a proper a torus P 3E. we a 2-sphere T N n to j set l E. so E one cannot int(N) follows the r standard 1-cell. H,(A) A can construct i n N w i t h 3P=D and is is a such iD=D. S i n c e =0. iE=E. Hence one shows t h e r e of see which E n . where u 3-cell generator small 3A=C iC i3E=3E i t 1 - c e l l of D and sufficiently punctured n is torus fixed 1 with a 3E and do not meet Fix s a d d l e p o i n t s we i (T-N) conjugate of 2-cell u bounds involution proper t near punctured transversal that annulus (T-N) u of M proper is 1 of an tT) U that i3D=3D, by a proper iP=P (namely isotope 2 T N). Consider n transversally 1-sided. under Let i. satisfying (IV). If the at S, torus and (I). If U Fix W be a r e g u l a r 3W i s T*=P D. 1 is 3W i s contained in T ' . neighborhood i n c o m p r e s s i b l e then compressible intersects Fix it we of T* T' is invariant i s a 2-sided arrive T' at torus case 52 We may now assume T iT and F i x 0 disjoint are T with pairwise 1-spheres successively consisting Therefore any any See Note: except regular are T is incompressible, have embedded n i T , T n i T iT n T tori or in also arising We bottles T n i T T and iT. a standard follows from neighborhood W with of 0 keep a l w a y s have It 0 iT . Klein always 2.9-.10. T , consists 0 and 0 annuli will bottle 0 in but satisfies instead T. If of We can n n i T iT, then a then such 3W a conditions 2-sided. is " a torus. a solid of W (I) be 3W and 2-sided. 3W i s solid Let W so t h a t 3W i s If the Fix 3W is compressible, by torus. torus If a If iT=T V contains an contradiction. main c a s e s and the take (I). Now V=M-W iT T=0 four T of n torus are all tW=W o r W i W = 0 . a torus. 1-sided 1-spheres annuli 3W s a t i s f i e s (IV). There of T 1-sided and T is and T components. applicable. transversal we of n e i g h b o r h o o d of Lemma 6.1 no s a d d l e bounding a regular Suppose that T Klein has is has Definitions 1-sided So Lemma 6.1 in 1-sphere construction o T incompressible 1-spheres neighborhood. that t bounding 1-spheres of n transversal construct fewer 0 the now d e p e n d i n g on t h e compressibility of number certain sur f a c e s . Assume T A £ into iT two be n i T an consists innermost annuli A 1 of at least annulus: and A" two Ar-|T=3A. with 1-spheres. Let 3A decomposes T T=A',,A n and Figure 3A=3A'=3A"=A' bottles. n A " . T ' = A ' A and T " = A " A a r e t o r i U See F i g u r e Case 1 ) T ' Case 8. is or U 8. incompressible. 1 . 1 ) t3A=3A and i A = A ' . Then cT'=T'. One sees is Fix transversal c o n s i d e r i n g t h e s t a n d a r d n e i g h b o r h o o d s o f 3A. We case (I) or Case single 1 . 2 ) Either i 3 A = 3 A a n d iA=A" U standard 7 , U 3A b e c a u s e 7 2 tS=S. int(V (7i[_j7 ) 2 so innermost so T , , of annuli " ) isotopic Proceed case neighborhoods distinct A arrive at or i 3 A 3 A n is a 1 - s p h e r e S and I A £ A " . standard U t o T ' by (IV). In t h e l a t t e r (A' Klein n 1 Let V, V ). Then T ^ i T , 2 (71u72)=0« with T , . and be 2 Fix 2 distinct be t h e two b o t h A and A ' . L e t T T, is incompressible. iT^ V 3A a n d l e t 7, and y t h a t meet 1 u and ' and T, L (T iT) n are Fix (7, 7 )=0 are n u pairwise 2 T, - ambient and A i s transversal. 54 Case £ A' Either t9A 9A is n a single 1-sphere S and IA o r i9A 9A=0. n Let let 1.3) 3A=S S'. 7 be t h e Ax[0,e] Let U V be a s t a n d a r d standard annulus be a s u f f i c i e n t l y that thin n e i g h b o r h o o d of meets collar of S and b o t h A and A ' . Let A=AxO in M such that S'x[0,e] A ' , C Sx[0,e] r-Tand ( A 3 V ) x [ 0 , e ] = (Ax[0,c3) n The c o l l a r case, tS = S assume (Axe) V other cases Then T and i since n set i T , it pairwise By L is By Lemma that both are the first (Axe) r-| 7*0 we may transversally in if transversal S)xL0,ej) u ((T tT)-9A) u n S,. to F i x . In a a 1-1 Proceed we may (Sx[0,e])-A' u T, isotopic transversal. l) 5.3 In Define A'-((S' u torus. intersect n ambient case a solid t((Axe) V) S,=S. = (Axe) V is 17=7. and S, T, since and n 1-sphere ( exists 3V. n is to T ' . with now T incompressible i T , and F i x 1 f are T,. assume T' and T" are A T=3A, both compressible. Case 2) For corresponding contains Then with every surfaces more t h a n let 9A^ = S 0 L annulus A two A, j i S T' h e r e with iT and T " a r e n c o m p r e s s i b l e and T n t T 1-spheres. and A w £ S 2 ' in s ' i T be a n n u l i a n < ^ s 2 a r e with A^ T=9A^ n 1-spheres and with 55 S,*S . Let 2 these A, 1-spheres i=1,2. See bounds is a T , =A trivializing ambient then, A, in since A . u torus then 1 u annuli and 1LJ of T that 9A^'=S A,' and A incompressible. the A U ' 2 U 0 u S^, Otherwise U. Say A,' A, A, ' u which on S , the A 0 was isotopic the trivializing be to A '|_jA 2 2 is ' |_, A compressible must a l s o 2 2 T, A,' bounds u A U. c i n c o m p r e s s i b l e T=A , ' A ambient 2 If bounds meet 2 bottle 5.5. A/LJA^LJA i n A ' A | _ j A . So T i s 1 is Lemma to A If three 9A=S S2 or a K l e i n by U T, 2 torus isotopic hypothesis. 2 9. u solid trivial be t h e A ' decompose T i n t o : Figure Define and A,' by torus trivial which was assumed c o m p r e s s i b l e . We have f i v e cases: Case 2.1) i (S, S ) = S , Case 2.2) i (S , u u 2 S ) =S , 2 u u S and 2 S and 2 Then Case 2.3) t S ^ S , and Case 2.4) i ( S , u S 2 ) n ( S , S 2 r. A. Then i(A ) 1 u A )=A. 2 cS r j A ' . 0 u 0 u 0 t(A, A )=A,' A ' tS =S . 0 iS Then = 0. 2 u tA,=A,' 2 and and tS 2 tS =S . 0 c A '. 2 0 56 Case 2.5) These i(S, y S ) 2 cases cover T^ w i t h fewer In case relabelling Then A 3 n 2.5) So By tS *S . we this a one have is the 1-sphere. In each case where i ( S u 0 s to we f i n d a 3 2.1) 3 ) n ( By 0 o u s s 3 ) = 0 A 3 a n ^ c a Let 2.3) s A be 3 with again we may assume 2.4) e the tT, c we may After intersection. tSo^So. case and 2.2) cases. the A,: S *S . other in we assume adjacent 3 from t h e 1-sphere 2.3) By c a s e 3 follows case annulus 3 is 2 possibilities. assume S, T , = 3A3 = S , | j S assume all S ) U 1-spheres. tS^S,. innermost (Si n tS *S . 0 gives 3 the reduct i o n . In the case 2.1) use T ^ t T , . Fix standard annulus meeting A, In case 2.2) and case 2.3): S^ with that meet In all = 0. Define isotopic T 2 to T, to given T Then T , tT 2 be 2 = T 1 7 U are tSo^S,. A i u A 2 = ( S x[0,e] o case Let A 1 U A i ^ A,. x First i n Define M T 2 - int(V (7i|_j7 ) 2 since be n the annuli u72) surface s 2 1 u ambient V ). 2 transversal assume u as c h t h a 2 tS *S . 0 t and (S,|_jS ). 0 By be a s u f f i c i e n t l y 2 0 7 pairwise (A! A )x[0,e] i u c 2.4): u let standard incompressible c T , j—j tT 1 - T2 n ' ^ 2 In i=1,2 the cases T, invariant. For 7^ to by and F i x 2 the transversal and A i s s t a n d a r d n e i g h b o r h o o d s of b o t h A and A ^ . is symmetry thin (S , S ) x [ 0 , e ] u 2 assume collar c of T and 57 (A, A )xe u Then T 2 2 iT n A-((S, S )xL0,eJ) u U ^ i(T iT)-S . 2 n incompressible T , . T , tT 2 If iS =S , 0 S x[0,c] and t(A, A )xe u Then T 2 i T n Case 2 ( i(T c 3) n t T surfaces is exactly Set tT=A. T= I A . , tT)-(S n two 1 u A 1 T' the c o l l a r (i=±l) pairwise 0 LJS,) ) if decomposes disc in V , . considering of S, we Next If J m and not then or J J D P D » P that standard (A 1 u A )xe 2 i n one 1-sphere iT with and T " a r e - n o t S . 3 A T=3A, both n i n c o m p r e s s i b l e and n n or K l e i n on t h e i r two a n n u l i and i|V, by S e c t i o n the standard u bottles interiors U. it and V l u bottle, follows 2 with 3V^=A^ tA^ are s o l i d K l e i n is a solid Klein This contradicts we show condition w i t h A . , A , = 3 A , = 3 A . , = T i T = S , S . Then say V , iV^V^ the 3 c that bottles. then S, since S, bounds a t h e i n c o m p r e s s i b i l i t y of T . By the standard a n n u l i see transversal. 1-spheres. disjoint 3V, i n t o to S . u and 3U^=A^ |_j t A _ ^ . None o f U ^ o r Otherwise, so transversally IA , . There are s o l i d t o r i U isotopic but r e p l a c e For each annulus A corresponding T intersect 2 ambient £ A . Use Lemma 5.3 on a to adjust 0 2 and F i x a r e p a i r w i s e 2 A , by S , x [ 0 , e ] n e i g h b o r h o o d of S u is 2 p r o c e e d as above 0 £ o T 0 ((S, S )x[0,e]-A). u 2 of a standard neighborhood I U ^ U ^ . and t | V . , 4, are o r i e n t a t i o n 1 1 i s conjugate involutions a r e not p o s s i b l e s i n c e S, on a or S to J solid 2 preserving. A , J rjf J 2 N f torus. would bound a d i s c 58 contradicting If the is 11V-, S | - ) F i x = 0 . Then also has only conjugate There is a band, 3W is arrive at generator taking is N M So F i x > contains V\ a r e solid tori W and By Lemma 6 . 1 , say £ Fix f i x e d set It and component follows t|V is 2 a Klein bottle of K with t9W=9W and K V\ is n is 3W S, a K. Mobius incompressible. We (I). t|V, is of H,(V,) tT t o conjugate 9U. , to and hence (move A , to J , then A an [S,] represents there is ambient iA,). This contradicts a isotopy that tT incompressible. suppose on D x S . = K x a 2 determine construct is for annuli A' extends J A of is tS,=S, and a conjugation a generator the t|V, If 1 conjugation iA' as for 9D xS . 2 1 2 1 above. j 9 n 1 If above. tS =S 1 but 2 the V, involution and 1 It is t a k i n g A, and we g e t 2 contradiction of to a c o n j u g a t i o n 2 to S , ' = 1 xS >9D xS 9V, H,(D xS ) case conjugate then S '=9D x-1 and p r o c e e d as above 2 then M neighborhood a torus. Finally J J T. boundary component. regular Since If J to of has a 2 - d i m e n s i o n a l 2 one to W Fix=0. n conjugate t |V 2 that incompressibility possible to A ' . >D xS'. 2 a Sj^-lxS But to This [S,'] contradiction as t h e n use S,'=9D x1 and this 2 time obtaining a 1 59 Case 4) Then let a a,, T i T tS=S. is Let 2 /3 /3 =0, 2 If T is 2-sided Tx[0,e] of int(a ). Hence 2 = (T = (T 2 t T u T=TxO u T, follows annuli cannot We a r r i v e (if T is If (I). since V is T-int(V) and a Klein either Assume at or T If M=V V U 1 U V with 2 - 2 and t j3 •, = /? . 2 int(V) int(V). 2-sided. Also tT^T,. sufficiently only thin one o f and 1-sided. are Since collar inMc^) therefore T, and "twisted" T, T are 2 and is S Klein is (if 2 c tori. b o t h of relative T is incompressible and T int(V), and with 2 ) - S (I). is 2 bounds a s o l i d t o r u s V^. or 2 0 ) or n e i t h e r T, torus a u separate that 1-sided then u a tT-int(V) T, a , of a s o l i d t o r u s and e i t h e r bottle) of 2 intersect From now on assume T i s This ta =a is that can u iT 0, u follows incompressible. S. standard annuli e 2-sided then T , it 1-sphere ia,=a,, 2 i n and T is be t ^ 2 Define T, T a single V be a s t a n d a r d n e i g h b o r h o o d 01 and 0 a, a =0, i n n a we to the V torus). arrive at are c o m p r e s s i b l e . Then T^ V\ then bottle, int(V,) a and contains contradiction. int(V ) 2 a So pairwise disjoint. By of a. the a, Therefore ba^. axr, of c h o i c e of standard that is the and a , t interchanges 2 t|a^ is conjugate involutions image o f of S'xO S'xI. under the t o one of Let S. components idxr, be a K X T or 1-sphere some c o n j u g a t i o n . 60 Note that does n o t bound a d i s c D in V , , otherwise T would be c o m p r e s s i b l e . Note a l s o 3A=S, S U as and 2 follows. the that Torus V, T =A B. 3 3(V V,)=T U If bottle, Since only if U S 2 Case conjugate to (I) i t|a can be given i s an a n n u l u s B. I-bundle we a r r i v e fixed set at i|a, at taking A (I) t | a , and i | a T T is meets fixed T T 3 is a 3 i f and conjugate. by in S, t | V , has a two fixed bounds a d i s c o r 11V, . By the above (IV). to a x r . Then i | 3V, i s c o n j u g a t e S^ t o S ' x t - I ) . on t h e b o u n d a r i e s . are 2 3V, that conjugate S.XK o r axk with 3 so If t o i d x T . Then follows or B [ V Consider over (IV). a (I). that 4 it (IV) interchanges does not s e p a r a t e i s conjugate 1 with and n reversing, a conjugation 3 an V, a standard neighborhood, F i x a , = 0 we a r r i v e orientation [ t h e components o f V, A. There T is a torus 3 By S e c t i o n 4.2) by A . S i n c e is Moreover bound an a n n u l u s comments, given V , U 2 4.1) 1-spheres. 1 T is 2-dimensional A at p r o p e r t y follows over F i x a = 0 . Therefore n annulus interchanges Lemma 6.1 g i v e s V Case S V a torus. 1-sided. Klein c i n V , so i t follows 2 an and iB=B. V i s an I - b u n d l e o v e r 2 It U is 3a,, I-bundle s t r u c t u r e w i t h 36=5, | | S is i s separated of A is trivial trivial there iA=A t h e n we a r r i v e components A-V,. if 1 In t h e f i r s t t o ax/c on S x S 1 By S e c t i o n by a c o n j u g a t i o n case i|V, 4 extending S, t|V, is 1 by is t h e one bounds a disc 61 and in the second case A g a i n by t h e above Case are 4.3) S^a,!-)/^ bound if 1 U S bound an a n n u l u s w i t h 2 comments we a r r i v e i s conjugate L\CI-\ orientation reversing S preserving. and only and w i t h o u t two a n n u l i if loss A , and A ways of i n d u c i n g an Now T say S, 2 and o n l y reversing if a torus. the o r i e n t a t i o n iA,=A the c l a i m 2 If i |9V T or If is t|V 2 with on Then i|V, is 2 i and and t | V orientation To s e e t h i s S,. S, let and t S , i A , = A . Consider the 2 iS,. The orientation i n d u c e d by a , a r e t h e same Since So if i|a, is orientation and only A, if orientation induce opposite T is a torus. Since follows. is a torus then to the taking S, t|V 2 is orientation involution ax/c t o i x S . As i n c a s e 1 on reversing S xS 1 1 so by 4 . 2 ) we a r r i v e a at (IV). T is a Klein fixed point t|V, tS, i s conjugate 2 conjugation (I) on . (IV). i n d u c e d by a , and t h e i n d u c e d by i a r e o p p o s i t e . orientations KXT or £ T. Orient of 9V orientation T is (I) is a torus. i n d u c e d by A , and t h e o r i e n t a t i o n if to at iA=A. free is conjugate to bottle so i | V J 2 2 t h e n we a r r i v e i s conjugate to j at g ( I I I ) . 119V or j ^ 2 while c QED IV. INVOLUTIONS ON ORIENTABLE I-BUNDLES OVER TORI AND KLEIN BOTTLES Involutions #7. As the on the Trivial an a p p l i c a t i o n involutions Definition Let of the on v a r i o u s I-Bundle Over annulus a Torus theorem we classify I-bundles. 7.1 W=S xS xI 1 be t h e 1 trivial I-bundle over the torus T=S xS . 1 1 Define the following k =idxidxT T k 2 A having k2g=idxKXT f i x e d set having =idxKxid having k = ( p . ( idx(c) ) x i d having k Q F torus two f i x e d set 1 four four the the 10): S xS xO 1 S x±1xl 1 1-spheres f i x e d set set Figure two a n n u l i f i x e d set fixed 4 on W (see the set f i x e d set k £=KX/<xid having k^p=/cx«XT h a v i n g fixed having A kg=(p.(idxK))xr involutions S x±1x0 1 annulus S x1xl 1-sphere S x1x0 1-cells points 1 1 ±1x±1xl ±1x±1x0 =axidxid k =axidx7 NI VTT k NF V7 =ax/<xid k =ax«XT Q I Here fixed p.(idxK)(z,w)=(zw,w). point orientation F means it free. The last four involutions The s u b s c r i p t 0 means t h e preserving, keeps t h e N means it is are involution orientation b o u n d a r y components f i x e d is reversing, (as sets), k 4P (•Ov^»(«C?^ 64 and I means it orientation the interchanges type of d i m e n s i o n s of involutions their are orientabi1ity the not type the two other components. involutions fixed point conjugate, and F/I. boundary is determined sets. because properties The These by eleven fixed point sets, conjugacy class are invar iants. and kg a r e define Let the identification k ' =( idxrxid)/d with s involutions can representations Lemma following. d=axT|gjXid. fix fix be the given 1 1 (S'xOxU/d (S xOxO)/d. and The 1 similar S xlxl Then W = S x l x l / d . . set set On alternate other conjugate u s i n g d = a x r | x i d or d = i d x r | x i d . 7.2 Let A be (Tx[-1,1],A) By Proof: A as an TX[-1,1]. where J an annulus in TX[-1,1], Then =(S XS X[-1,1],S'X1X[-1,1]). 1 1 isomorphism take show If is not an A - meets then let interval enough so t h a t TX[-1,1] essential one o f the boundary components S'xlxl-I,1]. First small to with A k '=(idxrxT)/d of conjugate A that A' chosen A' 3A=0 n meets Tx[-1,1]/(z,w,-1)~(z,w',-1), both be boundary the components annulus S XJX[-1,1] 1 sufficiently close and A ' the Tx-1. the is in Then, disc in to l e S 1 component the solid 1xS x[-1,1]/~ 1 of and of torus is a 65 nonseparating 2 - c e l l meeting A' i n one nonseparating of C. By Lemma 5.5 there i s a s o l i d A torus V that 1-cell trivializes and does not meet A'. N e c e s s a r i l y V does not meet Tx-1 so A i s also t r i v i a l in TX[-1,1]. This i s a contradiction. Next a d j u s t by an isomorphism so that A and 1xS x[-1,1] 1 meet i n a s i n g l e proper 1 - c e l l . Then the isomorphism can be constructed. QED Theorem 7.3 Let isomorphic free i and i ' be f i x e d p o i n t s e t s . I f t and ' assume, in addition, o r i e n t a t i o n type and that of involutions that on t 1 both W=S xS xI 1 are with 1 fixed have point the same c interchanges boundary components 3W i f and only i f t' interchanges boundary components of 9W. Then conjugate i and t' a r e conjugate. An i n v o l u t i o n on W i s t o one of the eleven i n v o l u t i o n s l i s t e d P r o o f : L e t i be an i n v o l u t i o n on W=S xS xI. We 1 conjugate 5.9 to 1 above. show i t is a standard i n v o l u t i o n . By the Annulus Theorem there i s an e s s e n t i a l annulus A with e i t h e r tA A=0 n 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 previous lemma take of A A of i s not interchanged. form S x 1 x [ - 1 , l ] . With 1 By the further 66 adjustment take i A = S x - 1 x [ - 1,1] W =S'x{x+yi:y>0}x[-l l] 4 Case are three Let 1 cases: iA=A, A a n d F i x a r e t r a n s v e r s a l 1) iA*A. and W . = S x { x + y i : y < 0 ) x [ - l j ] . ( There if 1 and t h e 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) i A A = 0 a n d iW =W . Case 3) iA A=0 a n d iW =W. . n n We show t h a t in case 2) k in case 3) k conjugate listed A' Q F N p Q F , * , k , k case. under. Each QF' k NI' k point k , k N p k it of 2 g 2 , or k k , Q I or k A k , ft k g 2) t a k e it to display S x1xl, 1 T' t a k S xixl; 1 1xS xI e i n case for 1 4 p . k 2 A ' in more (or at l e a s t i n the case suffices form or k 4 C are l i s t e d involution arise g . involutions standard involutions fixed simultaneously , k g, k has been an a n n u l u s with properties i n case 1) t a k e k a 2S a n 3) S'xlxl d t a k e A two free, 2 A , for k , kg. 2 isomorphic , 2 A of A . C o n s i d e r A ' as f o l l o w s : in case {(z,z ,t):z,t} Call N J k T To s e e t h i s to those IxS'xI; k Q I , k , of one) can i n f a c t analogous for m of t h e s t a n d a r d i n W, n o t n e c e s s a r i l y take o i i s conjugate t o : 1) k one + + in case Several than + if sets they on W o f same t y p e and, in a d d i t i o n , have interchange the o r do when if they they are same o r i e n t a t i o n not interchange have fixed type and boundary 67 components. type It suffices as a s t a n d a r d case, and into the same c a s e standard done are 1) - 3) is made of not t i n d u c e s an with S ' x 3 I x [ - 1,1 ] . Fix(X) in a five A also collar and of standard S generator. is same corresponding, and fall conjugate. 4. Reserve All for where proper the j to denote constructions t ' , even if not and t h e collar a 3Fix(j's)=0, j g the X.d=d.X when Fix See Remark 3.6. A^S'xl and where since and torus restricted to transversal Since By the to collar adjusting (Lemma 3A. The solid t | A and X|A a r e component of Let is in V. on an a n n u l u s conjugate then, on since H,(3V)=Z@Z [S]=[L]+a[M] not to 1 X 1 fixed 2 d=idx(T|31)xid. copy S X I X 1 X(A)=A. Write V=D xS . is conjugate a transversal property involutions be orientations X are A we may assume generate [L] are performed t has t h e same t y p e torus. involution the denote interchanged Let they the Section 1 1 not be under have solid that interchanged. V = S x l x [ - 1,1] is i' W=S xlx[- 1,1]/d involution Let on t h e to listed then iA=A, A and F i x 1) Then A. i and first, stated. Case A is use t explicitly of if involutions for show, involution second, Constant to one of c the 3.2).' [S]eH,(V)=Z L=1xS . is Then 1 with a proper choice a [M] of aeZ. to j jg is g X|A=axid. Therefore , J , M J D p or orientation S is kept J p 2 : J f ^ were preserving setwise and f i x e d by 68 X. So [S/X] However [Fix(X)] [Fix(X)/i] j represents is then w t # twice also = generator of H , ( V / X ) . I f i*[L]+ai»[M] = contradicting <.A=A. I f ~[L]-[M]+a[M] t ±[S] contradicting to J = 0, 2 P then since 2 p is X is of H,(V) [L]+[M]-a[M] to j ^ tA=A. I f orientation and X were This i*[S] = conjugate and implies to ±[S] * then p reversing [S]=i*[S]. conjugate component investigate see t h e s e select 1 9Fix(J ) 2 P [L]+a[M] = a contradiction. Hence a H (V/X)=Z. X were c o n j u g a t e X were c o n j u g a t e X|A=axT. T h e r e f o r e ~[L]+a[M], be j of represents a generator a generator [S] the a the give conjugate five D , J for involutions of to h:A >A' 2 u J is ^ x a two i has t h e n d or Q J meets X|A. between N Let > S. S i n c e we different to a B We will types, X|A and X ' | A ' x are in the same component of the same Fix so e x t e n d X'. A c o n j u g a t i o n h:W a r * e n t 1-spheres. So component of Fix(X|A)={x,y}. °-( ^ 1-sphere. it , j and conjugation d ' (A' ) . 32C' F i x ( X ) then F i x C that possibilities 1 . 1 ) X|A=KXT. L e t to 2 T h i s conjugation extends >A' u A u conjugation h:A d(A) J , J V-int(A d(A)) rise choose S ' = h ( S ) . Case of to type of k g ^ i e s a Necessarily m component of e Otherwise or Fix(X') k s' 2 iff N o w Fix x ' a n is d x and d ( x ) >W' between >V t and between i' is one d'(x') are F i x ( X ) . The c o n j u g a t i o n e x t e n d s t o a c o n j u g a t i o n h:V is X X induced. in over and 69 Case Fix(X|A) Thus i s two 2 - c e l l s , i has type since over 1.2) X|A=/<xid. P r o c e e d a s i n c a s e the annulus torus and and N 1.5) = -[L]-a[M] . Let path i n B^ f r o m 9J t o lifting an may a l s o assume t h a t arrange that property h to follows k [S] = N I I , u I 2 A XB B=0. X*[S] B^/t n u J u d ( J ) = to j U n d(9J). So n is J Q B) . with J X J = 0 l^ cl^=0. 2 C to J X is 1.1). 2 C or j *([Zj+a[M]) and Q t i s of type Let J be a and l e t 1^ be a n y an annulus s o by By making p r o p e r c h o i c e s , we bounds a 2 - c e l l in V. A similar Use I ^' and 1^' t o extend 9V > 9 V and c o m p l e t e t h e argument a s 2 ) i A A = 0 and tW,.=W . L e t S be a f i x e d component before. Case of n 9A. L e t B be a component o f X=t|W . + + W + n 9W T h e r e a r e two p o s s i b i l i t i e s : 2a) XB=B . = i s p a t h h o m o t o p i c t o I^/i we f o r X' f o r J ' = h ( J ) . conjugation so F i x i s A . Proceed as i n u embedded p a t h t h a t holds a It 2 of similarly T and B = 9V - ( A d ( A ) 1-cell B of k . Proceed as i n 1.1). - [ S ] , So X i s c o n j u g a t e nonseparating to J X | A = a x i d . Then X i s c o n j u g a t e B,=B over IA) . c i s of type i s not p o s s i b l e since c Q F U t has the type to J Case k A u extends two c o m p o n e n t s . E x t e n d 1 . 4 ) X|A=axr. conjugate 2 B into 9V-(B h A now two o r one a n n u l i . 1.3) X | A = i d x r . Then X i s c o n j u g a t e Case j o r k . Now iB=B. 2 A F i x separates Case a of k so F i x i s e i t h e r 1.1) e x c e p t that meets S. Let 70 2b) In 2a) 2b) one o f N interchanges b o u n d a r y components o f does n o t . is i it XB B=0. the W + standard This splits a solid torus, involution the of present the case 3W, so X i s solid into while in conjugate to torus. four subcases. In fact in case 2a) if if in case 2b) if i is orientation then X is i is orientation then X is i is orientation then X is c is orientation then X is if To show Since In S case this £ 3W -Fix, 2a) then is +1 i f In involutions satisfy X and X. i n of combining a J A j or to j to J 2 p [S] be c o n j u g a t e to If in case X is other N or g + while J » j Q . . is J 2 D 2b) a r J or M J p« D X*[S]=M(X)[S], preserving conjugate cases generator. and -1 to J if or 2 C X j 2 p This X[S]=[S]. claim. X . = i | W . must Combining to in H,(W )=Z, orientation all 2 reversing conjugate reversing. X[S]=-[S]. the X is J c* preserving conjugate that to reversing conjugate X*[S]=~M(X)[S], orientation establishes conjugate X cannot + where M ( X ) is note preserving all the the types as with a J < gives N (similar) different listed an case 2a) possible previously. involution - 2b). ways gives For of type example, k^. 71 Let are i' be of same t y p e conjugate. equivariance that It remains >A' y i ' A ' . u h extends to W to show It suffices to as a b o v e . 2.1) X i s conjugate to J A < Fix B n is a 1-sphere are i n t e r c h a n g e d . Extend h one and t h e n e x t e n d o v e r these equivariance. the show are conjugate. + and t h e components of B - F i x of they >A' and e x t e n d by when X=i|W+ and X' = t ' | W + Case i. F i n d an i s o m o r p h i s m h:A to h : A i A Take B and B' as components Similarly conjugation f o r the other a l l o f B by annulus of W e x t e n d a b l e p r o p e r t y of J over + n 9W. By this conjugation . Then B/t i s a Mobius a extends t o a l l of W . + Case band. 2.2) X i s conjugate The i s o m o r p h i s m h to J extends N to B/t. Lift to B and proceed as i n 2 . 1 . Case 2 . 3 ) Fix B/t X i s conjugate being n two extends the given points of choices points. n This to F i x ' n Case 2 . 5 ) 1-cell B / / 9B/i isomorphism ^ t s 2-cell a with i s an i s o m o r p h i s m w h i c h B ' / i ' X i s conjugate and in lifts takes either and the two of t h e two with correct t o J p - E x t e n d h i n any way t o 2 B a n d t h e n e x t e n d by e q u i v a r i a n c e be - C h extends as i n 2 . 1 . Case 2 . 4 ) J 2 There i n d u c e d one on Fix B/i p o s s i b l e ways. to J a proper I of X i s conjugate to j t o 9W . + Q or j g . Then B X B = 0 . L e t 1 - c e l l of A and l e t J ' = h ( J ) . B with 9I = 9 ( J X J ) A u n n Select and a proper consider 72 C=I XI |_j J u XJ. u C cannot bounds a d i s c before let generate as in W , 2 in 1 the using fact the p:H,(9W ) H (3W ), 1 path c l a s s + be g e n e r a t o r s [Ly] loss for j of I Q for g since [ (I j Then let that Since we , Q jg this by w i t h S and by [A*,] + a defined + as for 9W . + , u is even. suppose C/X and p [ L ] = 2 [ L , ] and p[L]=[L,]. Since This I, u is and XI any and can d e f i n e u 0. double it J |_j XJ event has c l a s s determine C,' in the double covered C,=C. covered follows 1-cell we o b t a i n is Then p[C]=b[Atf,] a contradiction be a p r o p e r C/X So y=0. L e t M is is = (b/2) [My ] C,= 1 , In assume o b v i o u s homomorphism. L e t y again 1-sphere. [L] Let , p [ C ] = M [ L , ] + 2 b [ M ]. s follows J)/X] u As torus. 1 is C , j For . + changing I Achieve W /X H (9W /X) p[M]=2[M,] For By beZ. Now W' if [S]=[L]+a[M] by c o n c a t e n a t i n g be t h e p[M]=[M,] j 1 and X*[S]=[S], >H!(9W /X) + 0 or in choices some a e Z . even Then [M] and 1 for + that disc M=S x1 and L = 1 x S . 1 where u i s altering by C i t a + + and f o r bound H , ( 9 W ) = Z © Z and w i t h c o r r e c t [C] = A i [ L ] b [ M ] Without may not = D xS , + classes and C + W be u s e d t o e x t e n d h s i n c e so b i s by C. (I J)/X since 9(I J)#0. U Then lifts U even to So AI=1. i n B w i t h I , 1 = 9 1 , = 91 n [C ]=d[j^] 1 a curve different same way C, u it as C , for with such some d e Z . [C,]=0eH,(W ). follows (i.e.) a + that for u s i n g I,' i' if 73 C, u s e s I,. Extend h over I (and then extends to a c o n j u g a t i o n C a s e 3 ) i A A = 0 and and i' is of extend hrALjiA S'=h(S). an Let h:B A u they B u tori The W >W '. be any to B'. S determined and 9A i' +U by B have is the h(9B-S)=9B'-S'. h|3B on extends the to annulus t o an i s o m o r p h i s m of Extend to W=W W. let meets S. i and type free. conjugation of we have same o r i e n t a t i o n extends h isomorphism a Since property) isomorphism necessarily + >A' component define (F/I the i A + a This fixed point n type have >B'. h:A c is component of W + 3 W t h a t Similarly same i n t e r c h a n g e Since Fix B be t h e annulus. Let equivariance >A'|_ji'A'. case). by c o n s t r u c t i o n . + same t y p e . by for tW =W. . Then n Suppose I, the solid equivariance. QED Corollary Let 7.4 W and Involutions on W i s i then if it conjugate to k Fix,=FixxO Fix,', the trivial to a conjugation satisfies meeting be conjugate component of for W where h(9Fix,). are T or A k conjugation on W >3W is conjugation and l e t Then torus. is If conjugate t to extendable condition: Let be a Fix,x[-1,1] be a b i c o l l a r of 9Fix,x[-1,l] is a extendable. c' following Fix,' over and 2 A h:3W Fix=Fix(i) such that k I-bundles a require bicollars component that h Fix, 3Fix,. of Similarly Fix'=Fix(t') extends to an 74 i s o m o r p h i sm h:9W Fix,x[-1,1 ] >9W u There are c o n j u g a t i o n s of u Fix,'x[-1,l]. 9W t h a t a r e not extendable! P r o o f : For k , T conjugate the fixed k Let to T < components. Then isomorphism taking T=S xS . 1 1 Clearly isomorphism W set W be + is + separates. the to + to TxO W + taking Tx1 i' h on W of one o f TX[0,1] and t h e i s o m o r p h i s m h:TxO on i closure isomorphic 9W -Fix Let the by an F i x t o Tx1 where >T'xO e x t e n d s to be T'x1. t o an Extend by equivar iance. For k , let A i, corollary. It h:3W Fix >9W' Fix'. u solid follows involution t interchanges condition conjugation and hence corollary. h:9W Fix, u the h^av Cutting copies It * in to a open of F i x in i t s . h|(9W|_jFix) property ' L follows >9W Fix,'. u l ^ ' boundary. is k extends Since J , n all e a S to The X on V w h i c h to J h, * a By the i n d u c e d from a so by t h e N extends on W e x t e n d i n g 11 an h for of conjugation for W . > 9 V . Now X i s c o n j u g a t e e t statement along F i x gives of F i x . S i m i l a r l y bicollar, extendable 2A' W as i n d u c e d by an i n v o l u t i o n induces a conjugation k be extends two c o p i e s on W i s these on conjugation For h u torus V having and n over V h. s t a a components t e m e n t o f conjugation of F i x and 3W-Fix are h:3W Fix h >3W Fix'. u Fix annuli decomposes W i n t o . into. 3 C . c >W' C' by L J u By t h e b i c o l l a r i n one of t h e two 3 - c e l l s Say h ( 3 C ) h:W C to equivariance a conjugation L e t C and tC be t h e two 3 - c e l l s u contained extends Then to the d e s i r e d that extend coning to condition h a that h(3C) is F i x ' decomposes W to an vertex isomorphism and extend by conjugation. QED Corollary 7.5 i If W=S xS xI 1 is an then 1 orientation W/1 is preserving involution i s o m o r p h i c t o one o f t h e on following spaces: W/k = 2 s 1-spheres W/kg D xS 2 a 1 with F i x / k 2 S two u n k n o t t e d (±1/2)XS , 1 = D xS 2 1-sphere {(e generator of 1 7 r i t a /2,e solid 2 , r l t torus w i t h F i x / k g one u n k n o t t e d ) :-1<t<1 } representing twice a H,(D xS ), 2 W/k^j an o r i e n t a b l e W/k = 4 C s o l i d torus S xl 2 1 twisted with F i x / k 4 C I-bundle over = {four a Klein bottle, points}xI. w/k = W . Q F Proof: i i s conjugate representations except for k o p , for to a standard the standard involution involutions. W/k = S x { x + y • i : 0 < y } x l / ( g g ' ) 1 u k. Use In a l l the cases where g i s an identification S x-1xl of S'xlxl and on k. depending 1 S x{x+yi:0<y}xO/(g g') 1 is u g' is For a an i d e n t i f i c a t i o n kg Mobius note band of that w i t h boundary Fix/k . c QED #8. Involutions on t h e Orientable I-Bundle Over a Klein Bottle Definition Let the 8.1 W = S x I x I / d be t h e orientable 1 Klein bottle S xlx0/d, 1 twisted where I-bundle over d=/cx( r | 31 ) X T . More explicitly, W=S x[-1,1]x[- 1,1]/(z,- 1 , t ) ~ ( z , 1 , - t ) . separating annulus, band. on 1 See F i g u r e S'xlxl induces with an Define _ 1 the X|.d=d.X| where k=X/d is a a n o n s e p a r a t i n g Mobius zxsxl. X| on An i n v o l u t i o n X denotes W. X^S^BIxl) Fix=Fix(k) = .X|))/d. following involutions k =idxidxr/d having k„ z=T i s 11. The I - f i b e r s a r e involution (Fix(X) Fix(d u whereas z = ±i =icxidxid/d having f i x e d set f i x e d set on W ( s e e a Klein Figure bottle 11): S'xIxO/d two M o b i u s bands ±1xlxl/d 77 k2g=Kxidxr/d having k =(-/c)xidxid/d A f i x e d set having two f i x e d set 1-spheres ±1xlx0/d an a n n u l u s ±ixlxl/d kg=(-K)xidxr/d having fixed set a k 2p=idxTxid/d fixed set an a n n u l u s A having two p o i n t s kg2Q idxrxr/d having fixed = two k ^=(-«)xTxid/d having having = a and (S'xOxI ± 1 x - 1 x 0 ) / d u and 1-cells ( S x O x O ± 1 x - 1 x l ) / d 1 fixed k^=axidxid/d f i x e d p o i n t ±ixlx0/d 1-sphere f i x e d set 2 ^2p (~K)xTxr/d set 1-sphere set free two u 1-cells ±ix0xl/d two p o i n t s ±ixOxO/d and orientation k =axidxr/d N fixed point free and orientation These eleven fixed point Recall point involutions sets that sets the are dimensional involutions W is orientable. fixed conjugate sets with reversing are types reversing because are their different. even d i m e n s i o n a l and t h a t orientation the fixed ones with preserving since 8.2 Let W = S x I x l / d be t h e 1 Then i f ambient i s o t o p y moving A so t h a t nonseparating A is or of an orientable bottle. is not orientability orientation odd Lemma or are preserving essential A is I-bundle over annulus of form ± i x l x l / d i f there a Klein is form S ' x - l x l / d A is separating. an if A Figure Fixed point S 2 x - l x l sets for the 11. standard involutions. 79 Proof: Remove components o f A , ( S x i x O ) / d . For 1 n details see [11]. QED Theorem 8.3 Let over i i arid a Klein and t' t' bottle, have fixed point free same o r i e n t a t i o n Then conjugate Proof: I-bundle W = S x I x l / d where d = « x ( T | 9 1 ) X T . Suppose 1 assume, fixed sets in addition, are conjugate. t o one of t h e e l e v e n The p r o o f is similar to the proof In t h e l a t t e r W.=S'x[-1,0]xl. In t h e c a s e is above. it iA A=0 n case o r iA=A a n d A and by Lemma 3 . 5 , assume t h e By t h e p r e v i o u s lemma take o f f o r m S x - 1 x l / d o r s e p a r a t i n g of 1 where iA=S xOxI. 1 W is nonseparating make listed on By t h e A n n u l u s Theorem 5.9 t h e r e be n the to A iA A=0 have i s conjugate of A i s not i n t e r c h a n g e d . form ± i x l x l / d . they o f Theorem 7 . 3 . L e t We show 1 collar to t and t' a r e involution involutions annulus A with e i t h e r transversal. that An on W = S x I x I / d . involution. an e s s e n t i a l and i f type. i a n d t' standard Fix on t h e o r i e n t a b l e isomorphic t be an i n v o l u t i o n a be i n v o l u t i o n s A Let is nonseparating W =S x [ 0, 1 ] x l + 1 and and 80 There are f i v e Case 1) iA=A, transversal cases: is A nonseparating, and the c o l l a r Fix iA A=0, Case 3) iAr-|A=0, A i s n o n s e p a r a t i n g a n d iW =W.. Case 4) tA=A, A i s s e p a r a t i n g , n A i s n o n s e p a r a t i n g a n d iW =W . t iA[-jA=0 and A i s 1) k , k in case 2) k , in case 4) k , k in case 5) k , k , conjugate listed R A 2 p R A 2 M , k 2 M k g k k A' Each S x(-1/2)xl/d; 1 ± i 7 r / 4 2 g 2 c , k k , k Q Q or k N 2 p or A 2 p 2 p k g or k 2 c , k , k s 2 c or k N involutions To see t h i s in. case in case it are l i s t e d involution arise more (or at l e a s t in the case suffices in i t has to display a been an a n n u l u s a n a l o g o u s t o t h o s e o f A. C o n s i d e r A' 1) take 4) take S'x-lxl/d; ±ixlxl/d; in in case case 2) take 5) take xlxl/d. It set) A k o f one) c a n i n f a c t follows: k , standard in W with p r o p e r t i e s as , , g 2 c , 2 g 3) does n o t a r i s e . case. under. transversal separating. S e v e r a l of the standard one are t i s conjugate t o : in case case A and F i x of A i s not i n t e r c h a n g e d . We show t h a t than + + Case 5) and t h a t are of A i s not i n t e r c h a n g e d . Case 2) and t h e c o l l a r e and A suffices type as t o show, a first, standard that t h a s t h e same involution listed (fixed under a 81 corresponding case, type and fall and s e c o n d , into the if i and same case t' 1) have - 5) the then same they are conjugate. Constant standard done use is involutions for i explicitly are on t h e to be Section 4. Reserve solid torus. performed All for j to denote constructions i', even if not Theorem 7 . 3 . The stated. Case Proceed 1) identification is Thus two the fixed sets. Then Fix X|A=idxr made of of as for Case now d = « x ( r | ) x T five either we have of instead possibilities When X|A=/<xid we have is 1) of for X|A give different Fix(X) is two 2-cells. two M o b i u s bands or Fix(X) is d=idx(T|)xid. an a n n u l u s . one a n n u l u s . Then F i x is a When Klein bottle. Case component W + n 9W 2) iA A=0, A n o n s e p a r a t i n g and n S=S x-1x-1 of 1 that meets 9A. S. Let Let B iW =W . + be t h e + X =i|W . + Select the component of are two + There + poss i b i 1 i t i e s : Similarly for B. 2a) X B =B 2b) X B + X (S)=S xOx-1. gives X.(S'x1x1)=S xOx-1. 2a). 1 if X + B n + The c o n j u g a c y + =0. of X + satisfies (X. | A ) . d = X | A + Therefore satisfies class t Suppose Since 1 Similarly + and X . = t | W . . Then + + case X is X. 2b) evaluating satisfies then also case X. case satisfies restricted by 2a). at S 2b). case the 82 orientation type of i. Up t o symmetry Case 2. 1 ) x there are four + i s conjugate to X. i s conjugate to Case 2. 2) *• i s c o n j u g a t e to X. j j i s conjugate to is conjugate to J i s conjugate to J Case 2. 4) x* i s c o n j u g a t e to J to J Case 2. 3) x + X. X. These for give this i s conjugate rise case. 2 c 2 c A 2 P N 2 P to i n v o l u t i o n s The c a s e cases: and and and and with f i x e d sets i s completed as c a s e as claimed in 2) Theorem 7.3. Case 3) iA A=0, A some choice n S=S xIxO with generator o f H,(W ) 1 but + d*[S]=-[S]. and This nonseparating of H,(W.). and iW =W. . orientation. Now Take + Then S is a t|(W. S)=d.i|(W n is a contradiction so t h i s case + n S) cannot arise. Case also 5) ± 7 r / / 4 and t h a t cA i s c o n t a i n e d that isomorphism, cA=e ^ and n separates components iA A=0 A we xIxl/d. and t U = U . 0 W It 0 2 with U into. assume Moreover, i n U =0, 2 if follows i n one A=e ^ ± U 0 n u i of By A and tA decompose W i n t o 0 2 decomposes may components U , U , , U tU,=U A separates. 7 f / / A ' the a two suitable ' xlxl/d and 4 three = that IA U solid 0 n u S i s a component of 2 = t tori A and 3A t h e n 83 [S]eH (U ) is generator. X=i|U 1 0 interchanges have the the generator showed Since sets as 2 Then s e l e c t an equivariance arguments for same f i x e d Suppose a h:A >A' To >U '. choice generate can integer. I of A. an a n n u l u s . There H,(3U) Similarly is -*N' which we h:A iA >A' u >U,'. h:W annuli h ^S' -'O O case. extend u by iA' . extends The to shows Extend to R fixed this claim >W is That we o b t a i n 0 >A' following A' choose be a p r o p e r components ^A' i|U show that Theorem 7 . 3 . c U between and A an a n n u l u s in twice U'. 1-spheres and [M] i s a h|A by 2 t and for a proper isomorphism in trivial i n 14,(1!). using 1-cell J 3U. generator M and L where in A meeting U', in >U'. [S]=a[M]+2[L] 1-cell the Then an i s o m o r p h i s m h:U be made so t h a t Let 2 a This 0 into 9A r e p r e s e n t s for t o an this, J H,(U ). falls U be a s o l i d t o r u s extends of both also h:U, twice case. Similarly [L] i' The 0 and of as obtaining a conjugation prove [M] and set h:A isomorphism Let H,(U). to i n Theorem 7.3 Suppose a component S o f of 2) conjugation 2) iA a generator X in case is on a s o l i d t o r u s A and conjugate and c o n c l u d i n g t h i s Claim: annuli isomorphism 0 [SleH^dJ,) involution for above. h:U equivariance the case t o an an is to conjugation extends as X i has listed and representing same s i t u a t i o n j p. is 0 disjoint boundaries argument i' a of 3U so "a" is that The an odd both boundary I'=h(I). B with B=3V-A 3J=3I. is Let 84 S ^ I L J J . since Then SpiS, itself is a point. leaving multiple for A deZ. isomorphism 2-cell, fixed Choose h:A J i s an which can (2d'-a')[>']. isomorphism changes generators so Since extended by to a given [S,]=d[M]+[L] Extend h|A int(B-J) is to an an open t o an i s o m o r p h i s m = 2[S,']-[S'] Hence h e x t e n d s t o h : U il iA=A, A i s s e p a r a t i n g , and t h e c o l l a r by that coning 3U a and a ' a r e odd a n d [M] a n d [M'] a r e we g e t h [M] = ±[M']. 4) of [S,] = h * ( 2 [ S , ] - [ S ]) (2d-a)h*[Al] Since " c " i s an odd i n t e g e r similarly. U be > 9 U ' . Then J' >A J'. u h Case There where o f [ S ] , So we may c h o o s e J some h:3U [ S , ] =b[A)]+c [ L ] >U' . A and F i x a r e t r a n s v e r s a l 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}xlxl/d + and W . = { z : z = x + y « i , x < 0 } x l x l / d . Then iW =W + component of involution on and + and + 3A. X =i|W + W. are is + j of a torus D x S . 2 choosing correct H,(3(D xS )) 2 W where 1 1 solid conjugate is odd. Since these for torus standard standard 1 [S]=a[Af] + 2[L] M=±1 and u d e p e n d s o n l y the a 1 X + * [ S ] = / u [ S ] where conditions to L e t S be a L e t M= S x 1 a n d L = 1 x S . orientations a tori. iA=A on it i|A. involutions Then e follows Checking on a s o l i d gives: u=1 a n d X M=-1 a n d X Similarly i s conjugate + + for to j ^ , J i s conjugate X.=t|W.. to J2 C s or J . M or J The c o l l a r D P . i s not i n t e r c h a n g e d so i 85 and t|A have the same o r i e n t a b i l i t y type. We obtain four cases: Case 4.1) X and X. + t|A Case 4.2) X is and X. + X X For case fixed set cases 4.1) given t' - 4.4) with have >W . + + first case n first B/t is to n to is a an a n n u l u s w i t h h/t and the conjugation j ^ and j 2 and c isomorphism c l a s s We w i l l also is so different sets. Therefore of to t, a there is a conjugation and t h e p r o o f is to a Klein Fix is i d x r or axr. will bottle while In in two M o b i u s b a n d s . band. It follows >3W '. + In the h:A u t o a c o n j u g a t i o n h:9W + property n + >3W '. In + of j M the the In the >A' second case 9(B/t)=(9A/t) (9Fix W )/t. extendable the an a n n u l u s . conjugate Mobius of show t h e isomorphic to that a c o n j u g a t i o n h:9W,. lift j g or axid. show t h e so F i x Fix A=0 c a s e B/t extends to h e x t e n d s o v e r W. t|A 1 and p to K X T . B= 3W,. - A . B i s 4.1) case j ^ > A ' . We show h e x t e n d s case F i x A = S second axr. nonisomorphic f i x e d f i x e d set be c o m p l e t e . L e t In to t|A. Similarly 1 and M >cxid. conjugate we w i l l c o n j u g a t i o n h:A h:W are determines to conjugate conjugate 4.1) i d x r or to conjugate i. [ A i s to J conjugate are and X. + to conjugate c|A i s Case 4.4) are and X. + conjugate conjugate t |A i s Case 4.3) are both shows Extend cases this 86 conjugation of extends In case A*. Let component 4.2) t o h:W let 3A has Fix.xl. Hence F i x Thus 3W >3W ' + J to D P case 4.3) we show j Theorem it Q is let we can an a n n u l u s and two >A' + 2-cells extend conjugation must for an u open it + Similarly to u 3Fix that a '. + are conjugation extendable S be a f i x e d component o f only is j y property of 1-cell in A that tl=0. n 1-cell J, tJ [S ]=±[M]. u in I |j t l and meets W /t + However, case First 2.5) of S double covers S/t Fix is two 1-spheres. I be a b o t h b o u n d a r y components of for with where J Similarly with a obvious an odd i n t e g e r . Compare is + the [ S/i ] =a [ M, ] + [ L, ] . L e t Similarly A. W / i a contradiction. occurs s where a i s ]+4[L,]. even, be + possible. p[S]=a[M a >H,(W /i) + p [ S ] = 2a[A4, ] + 2 [ L , ] and 0 two p:H,(W ) not follows U is is Fix a be A 9Fix A) with it to [S]=a[M]+ 2[L] 7.3. .So 0 so conjugation t Let homomorphism. S =J bicollared Since + A-Fix. i + u Fix . ±2 of under torus. I of component + solid that number 2-disc W . In so is and so by t h e + bicollar a fixed set (3Fix interchanged be fixed the - + Fix. the Extend 3W u the b o t h components o f + + denote intersection F i x x 1 meets Since + + follows points. + Fix x[-1,l] of annulus. Fix >W '. + I'=h(l). 3J,=3(I/i) is for a lift t'. There is proper A such a proper and [J, I]=±[M,]. of J, u Extend by h:A p Then 1 . >A' Let Then to h:A|_jJ h : A J u two >A u J' u then >A' u J ' |_j t ' J ' . t J 2-cells h:3W and that >3W . extendable are The Now equivariance 3W -(A + of j on [S ] 0 to cJ) c o n s i s t s J u i n t e r c h a n g e d under condition property by u of t . So h extends and conjugation the to show h e x t e n d s t o a c o n j u g a t i o n g on W. In and case Fix those 2 are + given point, already Fix on t h e + u are since also Fix' w\' n t is Fix .. two + n 3Fix . 2 A. the conjugation n 1-cells F i x is a 2-cell are extendable in the one X.. and one and h / t h / t can be e x t e n d e d o v e r >3W '. + 1 + to for , Fix,, 1 + Fixr-|W is the Since + same component o f A ' Fix be e x a c t l y Similarly o b t a i n a c o n j u g a t i o n h:3W in where 2 cannot c + Clearly Fix + Arguments s i m i l a r A • (3W -A)/t 2 n + Fix^ that h(Fix A)=Fix' h:W + u + boundary. the Fix, 1 - c e l l s of W . then two p o i n t s conjugation f i x e d set show t h a t of On l i f t i n g Fix A n 2 has + Say f i x e d set 1- s p h e r e 2- c e l l . X proper i=1,2. Then t h e given 4.4) + and- same component of property of J 2 C gives a >W '. + QED Corollary On 8.4 the involutions extendable. orientable I-bundle with 2-dimensional fixed over a sets are Klein bottle conjugation Proof: For Then c u t k the A 2 p open on t h e extendable property other are over cases a conjugation extends fixed set of solid similar the to and those over use torus for the fixed the conjugation involution the set. J2p* trivial T n e I-bundle torus. QED Corollary 8.5 If is t orientable t h e n W/t W/k is 2 D xS 2 {(e generator a solid 1 of a solid 1 n l t 2 c = Q Proof: 2 D xl 2 torus 1 1 with a Klein following Fix/k with ) : - 1<t<1} 2 S bottle spaces: two unknotted Fix/k g one unknotted representing twice a two 1 a 3-cell 2 an 2 plane with Fix/k 1 - c e l l s and one D xl/(a|3D )XT linked orientable with Fix/k ^,= ( ± l / 2 ) x l 2 two = s 2 c ±1/2x1 u 1-sphere, I-bundle 1-cell over a fibers, = W. i is representations from /2,e * 2 = projective W/k torus the over H,(D xS ), ((3/4)(z/|z|))x0 W/k 1 on an 1 = s 2 C I-bundle W = S x I x l / d involution ±1/2XS , 1-sphere W/k preserving i s o m o r p h i c t o one of D xS 1-spheres W/kg orientation twisted = 2 S an the conjugate for following the to a standard standard subspaces involution involutions. of W by k. W/k Use the arises identifications on 89 their for boundaries: (x+yi:0<y}xlxl k g , and S X [ 0 , 1 ] X I 1 for k s 2 c for and k k 2 C 2 g and k , Q S'xlx[0 1] F . QED 90 V. INVOLUTIONS ON ORIENTABLE AND ON UNIONS OF ORIENTABLE TORUS BUNDLES OVER A 1-SPHERE TWISTED I-BUNDLES OVER KLEIN BOTTLES £9. Involutions Let g:T d:T x-1 >T 2 >T x1 2 torus With be an 2 M by 2 is g:T g*:H,(T ) respect matrix Let basis of basis QeGL (Z). 2 2 1 [S,], 2 of H , ( T ) 2 is uniquely and S ^ S ' x l GL (Z). M is 1 and let Define the irreducible and g is 2 2 M(g) T =S xS incompressible 2-sided t o r u s . >T ' 2 to the different where g >H,(T ). 2 Then M 2 a nonseparating isotopy Sets i s o m o r p h i s m where M =T xI/d. g T x1 Fixed be d e f i n e d by d ( x , - 1) = ( g ( x ) , 1 ) . 2 bundle 1-Dimensional [S ] 2 The is 52=1x5'. g* 2 a conjugate orientable if to determined of H ^ T ) , matrix Up with Q and 1 is Then with given by a respect M(g) only by Q of if to a M(g), g is y orientation only preserving, when the g is det(M(g))=1. orientation p r e s e r v i n g when and 91 Of M 1 r interest '«',M . the g=id. 1 g=KXK M : g=cj. ( K x i d ) . p ( i . e . ) M„: g=cjc()cxid) M : g=w.p.(Kxid) 3 5 (i.e.) Then M(g)= M : 2 Each of t h e s e fixed orientable flat space See [ 1 5 ] . T h e s e a r e d e t e r m i n e d by g a s 5 M^S'xS'xS : J ?] • follows: . g (x , y ) = (x , y ) . Then M(g)= |^ ^ (i.e.) . g (x , y) = ( y , xy) . Then M(g)= g (x , y) = ( y , x) . (i.e.) spaces forms Then M(g)= g(x,y)=(y,xy). has Q j . Then M(g)= ^ involutions with } j. 1-dimensional sets. Let a are W, a n d W Klein union bottle. along I-bundles an irreducible. determine a Klein involutions k reverse signs the 2 M and k (0,1) A 2 p of isomorphism d determines o these W 2 annuli a to r e of is 1 annuli 2 o f W, up to o r M o b i u s bands o f W, s o m o r GL (Z). 2 Note Pbisms generators. a matrix S . H!(T )=Z©Z sign. i twisted M^ i s d o u b l e over nonseparating ( 1 , 0 ) of l 2 over determines a orientable bundle up n >9W u generator (nontrivial) I-bundles M^=(W, W )/d. The 2 a generator two torus T =3W,. a canonical of bottle orientable Let twisted i s o m o r p h i s m d:9W, boundaries Separating determine be two o r i e n t a b l e An the over c o v e r e d by sign. 2 As that of W, before, the that the 92 An alternate M = T xl/ and d + isomorphisms. twisted over are fixed point Then T xO that M a Klein is 2 The o r i e n t a b l e classified are only M M but finite. is M a also need two incompressible tori Lemma orientable twisted bundle known as I-bundles bottles. f o r m s have been equivalence and M . 6 there Define M 6 by (x,y,1)~(-x,-y,1) I-bundles over since a Klein H (M )=Z ©Z 1 6 2 the Hantzsche-Wendt 2 is manifold lemmas which i n M. F o r describe details see the p o s i t i o n of [11], 9.1 Let M be an o r i e n t a b l e incompressible torus If d:T x-1 2 If two forms M , , ' - * , ! ^ torus two twisted two K l e i n Up t o a f f i n e orientable reversing [4]). We be an M into (x,y,-1)~(-x,-y,-1), not is 6 orientation 3-dimensional space [15]). such space a u n i o n of bottle (see six + orientable 1 flat is bottle. S x±1xl are Wolf = S'xS'xI/ 6 is 6 (see a Klein spaces (d (x),1)~(x,1) free a u n i o n of bottle: these decomposes 2 I-bundles over Note for (d.(x),-1)~(x,-1) , 2 where d . description T is 2 T is is S 1 and l e t T then M = T x[-1,l]/d 2 where an i s o m o r p h i s m . separating orientable bundle over i n M. nonseparating >T x1 torus twisted t h e n T decomposes M i n t o W, I-bundles over a Klein and W , 2 bottle. M=W, W U Lemma and T = 9 W , = 9 W = W , W . 2 2 2 9.2 Let M be t h e a Klein bottle If T is bundle over If two lu 2 Proof: T be an I-bundles incompressible torus is an over i n M. orientable torus 1 s e p a r a t i n g t h e n T decomposes M i n t o W, twisted I-bundles 2 Two f o l d c o v e r on 1 n whether see 2 p 1 over a and Klein W , 2 bottle. . M by a t o r u s (T) t r a n s f o r m a t i o n of M i s details twisted S . and T=9W, = 9W =W W depending deck orientable n o n s e p a r a t i n g then M T is M=w W u n i o n of and l e t orientable For n is b u n d l e R. one or a fixed point A r g u e by two t o r i . free cases Note the involution. [11]. QED Lemma 9.3 Let is M be an o r i e n t a b l e a l s o a u n i o n of bottle. over S 1 basis Then so t h a t as M orientable has its its matrix a orientable matrix is bundle over twisted S . 1 is twisted 1 n a" , . £ Q Suppose M I-bundles over a b a s i s a s an o r i e n t a b l e T so t h a t torus and I-bundle over a torus a Klein bundle (canonical) a Klein bottle 94 Proof: M=T'xI/d p:T'xI >M bottle K. be some i d e n t i f i c a t i o n the both one components of the opposite is it On the I - b u n d l e s W, determines and a W of a Klein must q:R over M, as a u n i o n o f U^ = T x l restricted that to is Klein orientable torus has to the matrix for The' torus W, and W is also 2 must k^ Using t h i s = axKxr. the torus T' lift n T' 2 and W . 2 the twisted I-bundles over a fold two S , 1 deck free covering R=U, U U transformation involution W\ . k^ on U^ Hence k^ i n v o l u t i o n one s e e s | g such 2 is that ^J. 2 which d e t e r m i n e s n o n s e p a r a t i n g a n n u l i to two [-1 |_ 0 Abelianizing bundle H i s twisted W, W = T . we may i n t e r c h a n g e s b o u n d a r y components o f conjugate form assume The a fixed point Klein Since M orientable bundle over W\. A^ components H,(T'). the Klein a i n e a c h o f W, generators M K is bottle. annulus double covers 2 of a annuli boundary 7 and - 7 o f Let >Tx1. of Since a union of a b' kj. 1 0 is >M by a o r i e n t a b l e that have hand, M is 2 consists M has a m a t r i x nonseparating bottle, (K) 9(T'xI). Hence by c h o o s i n g a p p r o p r i a t e matrix 1 elements follows other p of A^ representing orientable d:T'x-1 induced p r o j e c t i o n . M contains I s o t o p e K so t h a t meeting bottle for the tori a l -1J 2 and t h e r e f o r e ["1 " L0 a matrix for of R -2a" 1_ f u n d a m e n t a l g r o u p of a torus bundle M c 95 with the M -c matrix . 1 c , one s e e s t h a t 1 0' c o n j u g a t e s 0 -1 isomorphic, matrix are It H,(M )=Z©Z@(Z/cZ). follows b=±a and Also c V 0 1 -c 0 1 to [ i so b=a f o r and M and c a suitable choice of generators. QED Proposition Let 9.4 M be an o r i e n t a b l e of orientable be an twisted involution incompressible nonseparating t o r u s bundle over I-bundles over on M torus and let with S a Klein T incompressible torus with a Then n or a union bottle. be T iT=0. 1 Let t separating there iT=T and F i x is a and T transversal. Proof: W, By Lemmas 9.1 and W ' are 2 bottle. and 9 . 2 , orientable Without loss I ^ W ^ L J W V twisted say iT u W iW,=W 3A =S , 0 u (T'x[-1,1]) I-bundles over Let 1 U U a Klein in W ' . 2 with 2 = 3W t(T'x[-1,1])=T'x[-1,1] 2 l twisted 3A=S i n I-bundles is a g a i n we see M = W , ( T ' x [ - 1 , 1 ] ) W with W i There 2 iS . 2 is Then o n i3A =0. o U s i n g the lemmas i=1,2) 1 i orientable bottle. an e s s e n t i a l 3A Klein W A be a n o n s e p a r a t i n g p r o p e r a n n u l u s S . where 2 a (for with W '=T over = T'x(-I) and i n annulus A By the 0 in W,. Write in T ' x [ 0 , l ] Partial with Annulus Theorem transversal iA =A i i there 5.6 t o Fix (i=1,2) Let T , = A must meet a torus. disjoint with or u are 9A o n i9A essential = 9A 0 1 3A U "twisted" and w i t h 2 2 either 2 t A u A T u follows relative iT!=T! A , and A iA,=A . A 2 . Essential annuli b o t h b o u n d a r y components so T , This annuli and since A is to T ' X [ - 1 , 1 ] T, is is of connected. T, nonseparating as is T'X[-1,1] i n W, is so A is T, is tA. transversal to Fix. n o n s e p a r a t i n g hence i n c o m p r e s s i b l e . QED Recall preserving - five and up t o c o n j u g a c y involutions I2C orientation preserving k^j trivial on orientation and that k^' on a preserving an there a n •'o d o involutions I-bundle involutions orientable are W k 2 twisted S n three a s o ^ i d orientation torus k s' ^40' 2 over ' ' k V, a torus s ' ' S2c'' k I-bundle over k and k a OF five 2c' Klein bottle. By following applying theorem. the T o r u s Theorem 6.2 t o M we o b t a i n the Theorem 9.5 Let of M be an o r i e n t a b l e orientable twisted I-bundles over be an i n v o l u t i o n w i t h a Then one of 1) There iT T=0. is over 2 is cT=T and i t s a trivial between 3) M There is Klein conjugate V o r 4) is k M there not component. torus T conjugate is with a o i is ^01" r torus T with ^2c' from r a an to t|W,' ° is an i s o m o r p h i s m induced I - b u n d l e s W,' 2 r o incompressible torus T with W ' W '=T. ^520' to ^ s ' t o r u s and d i s conjugate n I - b u n d l e s W, (incompressible) orientable 1 with i n t e r c h a n g e d . M = W/d where W over u n i o n of satisfies one is conjugate that i|W\ a separating to set i r e and and W ' over 2 i|W ' iT=T. are 2 ^oth conjugate a both to k 2s'' o'- exactly Proof: is on W t h a t bottle union bottle'. Let (incompressible) a nonseparating I-bundle the or a 1 iT decompose M i n t o two t r i v i a l collar is a Klein fixed t h e b o u n d a r y components of W. involution S following holds: a torus with There 2) 1-dimensional a nonseparating T and n and W the t o r u s bundle over case is (III) a (III) 1-sphere. to Apply case j Q . M is the of T o r u s Theorem 6 . 2 t|V, is conjugate to and 2 C i|V 2 6 T o r u s Theorem 6 . 2 . T Fix(t) M . Assume f o r does not o c c u r . Then s i n c e M torus J and satisfying case (I) is or t h e moment orientable (II) of that theorem. iT=T (1) By and Proposition the case proof, case (II) can (III) is Suppose 1 L jA. is 1 Lemmas case 9.1 and 1 if iT=T t h e n b o u n d a r y of be a collar eliminated. with (II) 9.2 a Klein iK=K. occurs incompressible torus that U., V, is U bottle. I A , is of K an invariant under 2 assume annulus bottle H,(V,). U. 1 U V,. H , ( 3 V ) = Z @ Z where assume p[M] = 2[M ] and of [M] 1 1 p[Z-] = [ L ] 1 H,(3V,/i). contradicts that be S, Then >S,/i Since follows u the by twisted a boundary so Klein of incompressible 1-dimensional S, the to a torus istrivial some twisted twice generators in H,(V,), integer generators 1 an i s o m o r p h i s m . iA, I-bundle a. homomorphism. suitable set As above p[S,]=2a[Atf ]+2[L,] is fixed represents obvious for jg. appropriate for 1 >H (9V /i) y So [S ]=a[M]+2[L] prr^OV,) it K=A iA orientable Selecting 1 iS^S,. an o r i e n t a b l e conjugate in the and [L,] i has a t |V•, i s of may the (I). Since generator [L] of is t g i v i n g case iS!=S . a Klein show is neighborhood over We U.i. regular separating case annulus 6.1 a at a separating Lemma is T. suppose by and F i x p - 1 5 ^ 0 , of If Then t o c o m p l e t e and Then Suppose separates. handled. bounds a p r o p e r M o b i u s band A of bottle T i n t e r c h a n g e d t h e n we a r r i v e a separating I-bundle over S is by c o n s i d e r i n g t h e that A collar 9.4, but [A^,] a [M] we Let Then and this 99 Now c o n s i d e r c a s e component solid S, torus and V ". M-(T cT) that u Then 2 W, be a incompressible orientable Let o b t a i n e d from V , " S . Let 2 of (III). V,' be t h e c l o s u r e contains y torus. 2 neighborhood By Lemmas 9.1 I-bundle over of a Klein 9(iT-W,)=C, It follows represent both is to the jg and t 2 to J 2 for 2 Q and for t case because S, 2 i, in V , ' / i , is meets only one 2 . be t h e shows its of meet lift a 9W, is W =M-W, 2 an is an iT-int(W,) See F i g u r e 2 to j is not g or 12. in 2 conjugate u 2 components would be a generator 6.2, j ^ . The 9V -(S, S ). both component of and 2 i n v o l u t i o n on V , ' t S! the .copies twice a generator conjugate (II) cannot otherwise C be 1 From t h e T o r u s Theorem i . r e p r e s e n t i n g an odd m u l t i p l e Hence Let 2 u the for V ' bottle. i n t e r c h a n g e s c o m p o n e n t s of 1 - c e l l S,/*, Fix(t,) 2 H,(V '). g i v e n above since and S Similarly i. conjugate argument S, 2 H,(V,') i n d u c e d by i, C , V, of two T. and 9 . 2 , a s e p a r a t i n g a n n u l u s . Let 1 f Let Similarly is C V,. by r e p l a c i n g S by = V,"/(S ~S ). regular twisted V," Also of 1-sphere of H^V,'). Fix(t,)'. Therefore 1 Fix(i) is one 1-sphere. Since annulus W 2 of n that a Let C t i D|-|D=0. 2 of H,(V ') 2 Also t A i (3D i A) n 2 1-cells V . 2 separating of nonseparating annulus W, and a of and 3D choice of the to there r to j 2 that (i,i )|iT 2 C 1 f c is iT|-|W of 2 (0,1)-generator A, one of T. twice . a 2-cell V ' A n in is v n u u two and t h a t D proper s in V ' . Since i, multiplication by 2 bounds a d i s c so 3D bounds a d i s c is proper nonseparating 1 A) (3D A) 2 two C,, with 2 Then ^ i ^ ^ H D of 1-cells. an a n n u l u s o by represents which disjoint A as i,t (3D n determined 2 meets 2 (DyD, ) W, in nonseparating (J rj)* = id transversally of V ' a proper four 2 S 3A D proper and and that of A are n annulus j^ A . Now c o n s i d e r follows But be 1-cells two conjugate it in T meeting arrange in 2 2 A conjugate nonseparating meets 1-sphere Let is 2 generator -1, the (1,0)-generator be a 2 point. Since is W, and of the correspond. 2 one a b o u n d a r y component o f of tT W, we s e e W is Mobius D, in V , ' . bands so in By its 101 boundary i s a (0,1)-generator (D D )r-|W u 1 is 2 a of" W , . On the n o n s e p a r a t i n g annulus of W two b o u n d a r y components a n d does not other since 2 separate hand i t has V,'. This gives M . 6 To show how M arises 6 in this 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 A Let ={(e j i,=JQ=idxa ( e e 4ffit) 9Aj. Let Vj"=Vj'/d. the identity _ 2 give 1 the and : t e R , 0<V<1/4} 27ri(l-t) Define h:A e identification 4*i((3/4)-t)) >A, t o be i n d u c e d from 2 and d e f i n e h|(t A ) 2 h:9V ' = i .h.t |A 2 1 2 = KX(-K) 2 >9V,' i s h ( x ) = x i f x e A , and h ( x ) = ( K X - K ) ( X ) 2 otherwise. has ( e we 13. L e t V j ' = D x S L e t d be t h e 2 27rit ) 4 7 r i t and I = J 2 £ = K X K . on (ie) ,e 2 7 r i ( t + v ) manner Then M 6 =V 1 ' U V f i x e d s e t one 1 - s p h e r e . 2 ' / h (In u the d and i n v o l u t i o n previous t , u i 2 construction one c a n t a k e D D ' = { t = - 1 / B } ) u QED C o r o l l a r y 9.6 Let over S Klein set 1 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 or a union of o r i e n t a b l e bottle twisted with a 1-dimensional i s o n e , two, t h r e e or four torus I-bundles bundle over a f i x e d s e t . Then t h e f i x e d 1-spheres. 1 Corollary 9.7 Let M an o r i e n t a b l e orientable an twisted union over of is a lens is In 2) 1). 4) k the 3) For In set. 3 or the 3 Corollary t | i s t be boundary twisted 8.5. conjugate b o u n d a r i e s of to k w s 2 i s lens for k c' /k V/i a 3-cell solid fixed P #P , I-bundle Consider k^ t h e n M/t T to k 2 s or two s o l i d t o r i . w i t h an o r i e n t a b l e / 2C w and c a p p i n g is and when e a c h of conjugate gives 2 3 Let the is the kg, When one boundary twisted M/t union I-bundle- o v e r a bottle. In in 1) a s o l i d torus S X1 P , bottle. Theorem 9 . 5 . t | i s Klein space, 7.5 a u n i o n a l o n g the of 1-dimensional or a u n i o n of 1 bottle. P r o o f : Use C o r o l l a r y In a Klein S a s o l i d t o r u s w i t h an o r i e n t a b l e a Klein c a s e s of bundle over I-bundles over i n v o l u t i o n on M w i t h a Then M/c of torus 2 s ', and k 2 c ' is 2 x I * space 2 c is n e identification of S x-1 2 with S xS . 2 1 and k^' ', we get capping W/kg ' 2 c the same s p a c e s gives a as 3-sphere P . 3 a 3-cell U T kg' gives so V V,/L t o r u s M/i s is a lens and V V , / t n is a solid torus. two 2-cells. Since V 2 is V,/i also a space. QED 103 Note that torus I-bundles over a Klein In the following subgroup K of GL (Z) 2 Note Z if i >GL ( Z ) 2 ?] •"<> 2 be [I in the ?]• sequence: >GL ( Z ) 2 twisted 9.3) i n d u c e s an e x a c t M be an o r i e n t a b l e twisted involution and o n l y a) the by of EeGL (Z) 2 generated >K Lemma Q e G L ( Z ) . Let 2 orientable h a s an let (See be u n i o n s >1 2 9.8 Let of bottle. >Z =Z/2Z 1 Theorem b u n d l e s may a l s o if M is a torus following for 1 0 0 -1 (1 ) 1 0' 0 -1 (2) Q-1' [ (3) Q" [ (4) 0 -1 E P where (5) E P where p (6) E P where p 1 1 bundle over I-bundles over f with one of torus f i x e d set a), b) or c) exactly below bundle with matrix some Q o r n a union bottle. Then M 1 or 1-spheres (n>0) holds. conjugate to one of E: n = •[*-?] 2 or 4 n = 1 or 3 -r p a Klein S n = 2 ?] n = 4 = [ J 1 n = 1 or 3 = [ J ? = [ rj o r [ 1 2] n = 2 104 b) Klein M is bottle canonical In (7) the a u n i o n of with field the of integral. (7) 1 0 0" -1 Q 2_ (8) 2 0 0" 1_ (9) 1 0 0 2_ 1 -1 -1 respect but to following of m a t r i c e s are I-bundles over some for taken is required Q 1 0 0" 2_ n = 1 or 2 Q 2 0 0" 1_ n = 1 or 2 P = (11) EP where P = (12) EP where P = (13) EP where P = M is M 6 1 0 1 1 1 Mi !] ]•[! iM? M i J] 2 3 0 E: over n = 2, where a of some Q o r t o be the p r o d u c t m a t r i x set 0" 2 EP to 3 or 4 n = 2 or 4 n = 1 or 3 n = 3 ] ~ [-? -! n = 2 and n = 1 9.9 The involution proof shows on M a r i s e s bundles over bottle. the twisted 1 0 (10) Remark (with one of inverses fationals be c) matrix generators) - (9) orientable S 1 how in each case (1) - (13) from i n v o l u t i o n s on o r i e n t a b l e or o r i e n t a b l e twisted I-bundles over the torus a Klein P r o o f : A p p l y Theorem 9 . 5 . Suppose case nonseparating torus Let T and W, Let or k with identification of these respect from S 1 involutions l^s' involutions = 2 H^S'xS ) The matrix of (2) torus of Theorem with 9.5 iT=T effect orientations are the m a t r i x k (1) Suppose c a s e an = Z©Z w i t h 1 where 1 the induce the 2 H,(T ) be 2 2 is 2 2 >T x1 C . Then M = T x l / q where c k„ nonseparating be 2 and [ 1 X S ] 2 and of >T xI 2 >T x-1. Consider in 1 h :W d:T x1 Then t h e 2 [S xl] is a M d e t e r m i n e d by standard Let 2 i is M(d)~ M(t )M(d)M(i,). 1 and t h e 1 .d.t,:T x-1 to basis and 2 h (T)=T x1. isomorphisms induced >T xI i|W^ 2 o c c u r s : There two components of i n d u c e d by T . identification 9.5 n h,:W, between Q I Theorem iT T=0. be t h e 2 conjugations kg with and W iT. of (1) and 9 C - (3) occurs: of k is above. There and the c o l l a r q is of T i s a not interchanged. Cut M open on T . is an i s o m o r p h i s m , that satisfies a conjugation Then M i is = Txl/d' X and k 4 C = the T xl/d induced involution k £=KX/cxid 4 matrix of on on 2 T xl. 2 The d:T =T x-1 2 2 T x l / d and 2 matrix from of M i s >T x1=T . 2 h:TxI >T xI 2 2 Then M 1 Let 2 Note i is the Txl be Define conjugate to involution conjugate d >Tx1 X on (with h(Tx-1)=T x-1). d=h|.d'.(h|T x-1)~ . 2 d':Tx-1 i n d u c e d from an i n v o l u t i o n d'.(X|Tx-1)=(X|Tx1).d'. between where to the satisfies d.(KXK) = (KXK).d. Now T / K X K is 2 of T /KXK, up 2 fixed and to is a twist "same" of by on S x i 1 if that see together twist d keeps a l l four then the of keep t h e [2], A twist has to all isomorphisms four p o i n t s ±1x±1 a twist with a twist With respect lifted 1-spheres) The c l a s s two Dehn t w i s t s : on i x S V ^ X K , the 2 four isotopy, direction. H,(T ) Hence to generated a twist a 2-sphere. the on S'xi//cXK on S'xi/zcx/c on S x-i in 1 basis lifts [S'xl], the [ixS ] 1 matrix points matrix fixed of (and d is in therefore the Fix is subgroup K of GL (Z). 2 In ±1x±1. general Label 4=-1x-1. number in fact and of orbits Note the induces ^ . induces of the that permutation (13). A l s o note K X K and have permutation (12)(34), (14 ) (23 ) . the (24) same m a t r i x and the following matrix the identity: inducing KXK, p, is matrix has and on commutes 3 that idxa same p (z,w)=(z,zw) permutation commute w i t h inducing (12). the jj. the d with £ Q and t is by commutes has points 3=1x-1 induced and has m a t r i x 2 has 2=-1x1, permutation (34) four f i x e d components of p (z,w)=(-zw,w) the on t h e 1=1x1, p , ( z , w ) = (zw , w ) p„(z,w)=(z,-zw) permutation as 1-sphere permutation the a permutation points a Dehn t w i s t . with K X K , | these The number of {1,2,3,4}. induces d induces matrix induces the isomorphisms axid (13)(24) inducing and axa 107 There that is a composition r d=d'.r and permutation: Use possibilities (243), generate E is form l i s t e d (4) - Suppose c a s e separating torus Then M i s W, and W W^ , of of with Klein As a b o v e , Then H,(3W )=Z@Z in 1 the (12)(34), Then use t h e and hence is of d is p's of to form i n K and P i s of , 9.5 occurs: There is a iT=T. two o r i e n t a b l e bottles. the nonseparating annulus Select 1 canonical of W, d:3W, >3W be 2 a i =t|W 2 I-bundles the conjugation are 2 standard representatives way: and twisted t | b y and 11 = 11W, i d=di . 2 Let by c h a n g i n g we may assume t h a t involutions. (14), So t h e m a t r i x d' identity reduce (24), (1243). Theorem a u n i o n of identification. on and to axa (34), the such (6). (3) over 2 (1234) isomorphisms induces and permutations. in d' idxa the m a t r i x these that permutations (1423), these EP where axid, to (234), such of (1,0) (0,1) arising from a of from a separating annulus. Suppose torus. We has m a t r i x and T . 2 the can d:3W, select Q 2 matrix fixed a basis if point for free. T, 1, i s T^SV^/i, such that conjugate to is p,:3W, k s' a >T, o r 2 ^s' ^ >3W 1 is M(p,)= ,„[| matrix p „d=q.p . 2 t 19W! induces Then t h e m a t r i x M(p )~ QM(p,) 2 1 an of isomorphism d is where (as Q is q:T, a product the >T with 2 in GL (Q) matrix 2 of ) q. 108 Conversely, up to isomorphism q:T, isotopy >T This 1 2 2 t|9W, Suppose conjugate or to k S 2 S fixed 2 c ' 1-cells boundary 2 k = 1x1x1 J J 1 x - 1 x 1 four of k ^' matching on S can determine the Proceed nonseparating arise arise k S2c' a n d 3 as m a t c is to use k £ ' = (-/cjxrxr/d 2 the representation arranged of meets 2 and represents labels labels occur kg 2 c ' combining a iT=T. k k ,' 2( 2 c S i n in the ' and 2 of i s , is ^2c' 1-cells is ) . : and invariant, 3 and 4 The same way curves to the fixed sets different The way. i f a matching The twist and d Fix. above In S meets ( 0 , 1 ) . The 2, in a case. case a 1, 1 to components torus with fixed u (e.g. the both S =ix1xi - i x - 1 x i to i n the in is invariant, For these n combining k t|9W, Then nonseparating annulus. assign n number from from to Fix 9W, i s performed 2 (13) be a 1-cell way kg2c' and 2 i f that k^^': fixed of lifts (1,0)eH,(9W,), is invariant, give a points conjugate s2 of one 2 free. convenience an (9). represents meets S - and (0,1). For and (7) k ^,' : S ^ S ' x O x l component only isomorphism and represents S, For determines 2 point are 2 QeGL (Z) instead. 2 both fixed t I , and for kg ^' For a KXK. this gives i s not £'=idxrxT/d. Kxrxid/d both to and 2 M(p )~ QM(p,)eGL (Z). a matrix the where above and a a k^' k g 2 c listing '. or T (11) from was a (12) and and (12) combining a S2c' QED 1 Let M be i. involution fibration xeS 1 not an orientable t Call >S p:M 1 torus fiber over preserving i(p (x))=p such that and i f F i x = F i x ( i ) bundle 1 if 1 Note the i n v o l u t i o n t(x,y,z)=(y,x,-z) fiber in preserving, this sense, with 1 there (i(x)) i s t r a n s v e r s a l to each S' is for a l l fiber p ( x ) . 1 on S^S'xS with respect obtained by p r o j e c t i o n to the l a s t c o o r d i n a t e . Corollary 9.10 to M be an o r i e n t a b l e t o r u s bundle over S . Then M has 1 a fiber preserving 1-spheres is 1 fibration Let a involution with fixed set exactly n (n>0) i f and only i f the matrix of M i s conjugate to one of ( 4 ) , (5) or (6) of Theorem P r o o f : L e t xeS 1 9.8. be such that the f i b e r Then T=p 1 (x) meets fixed set. collar of T i s not i n t e r c h a n g e d . So T s a t i s f i e s case Theorem 9.5. Now the cT=T. Since T and F i x are transversal- the f o l l o w the proof of Theorem (2) of 9.8. QED 1 10 #10. Involutions In were the On a 2 T previous defined. following With 2-Dimensional F i x e d section Recall involutions the also space Sets forms M , , Definition on t h e s e M 3.1. and 2 Define M 6 the spaces. M,: =idxKxid/d a =a)xid/d T having having f i x e d set f i x e d set the two tori S x±1xl/d 1 nonseparat ing-torus {(z,z)|zeC}xI/d On M : 2 /3 =idx/cxid/d h a v i n g fixed set two K l e i n f i x e d set the Klein 2K /3 =wxid/d h a v i n g K /3 =idx(-K)xid/d having T /3 p=KX/cxr/d having T4 f i x e d set f i x e d set bottles bottle separating a torus S'xilxl/d {(z , z ) | z e C } x I / d torus and f o u r S'xlixl/d points (S'xS'x-l ±1x±1x0)/d u On 7 K 2 p M : 6 =(-K>xaxid/d having f i x e d set a Klein two p o i n t s Here d denotes space form M . . the identification bottle and (S xS x1 1 x ± 1 x - 1 ) / d 1 for 1 u the c o r r e s p o n d i n g 111 Lemma 10.1 Let of be M be an o r i e n t a b l e orientable an involution Then t h e r e and is bundle over I-bundles over with f i x e d set a separating S 1 or a a Klein bottle. containing a Klein incompressible torus union Let i bottle. T with iT=T T Fix=0. n By Proof: of twisted torus Lemma 6.1 the K l e i n fixed bottle under invariant t, under T be t h e the K is b o u n d a r y of an arrange t and t h a t boundary of this a regular incompressible that the torus. regular i s meets F i x o n l y regular neighborhood K is neighborhood is at K. Since Then let neighborhood. QED Theorem 10.2 Let of be M be an o r i e n t a b l e orientable an twisted involution component. conjugate t o one of P r o o f : We a p p l y M F. set When F i s with a a Klein the seven contains a torus, an l f M 2 involutions the T o r u s Theorem S 1 or a bottle. 2-dimensional isomorphic to M Some economy c o u l d be fixed bundle over I-bundles over on Then M i s torus or M union Let fixed and 6 i defined above. showing that t set is 6.2. acheived incompressible by torus F c o u l d then be u s e d or K l e i n to the bottle construct a 1 12 torus that iT T=0. T with The n present i t g e n e r a l i z e d t o the involutions. M arise and and (III) case parallels is orientable d o e s not (II) It also so c a s e 2-sided do approach not of the has the orientation proof (IV) of of advantage reversing Theorem the Torus 7.3. Theorem Klein bottles do not arise since t is occur. So there i s an incompressible torus T with T iT=0 iT=T and T and Fix By or P r o p o s i t i o n 9.4 1) i T T = 0 and Case 2) iT=T, T d o e s n o t not T does not n either transversal. there are Case Cases orientation reversing. n 6.2 three cases: separate. separate and the bicollar of T i s interchanged. Case 3) By Klein iT=T and the last bottle We show T separates. lemma we t i s conjugate case 1) a 2 T in case 2) T a than one case it 3) case of case. is 0 does not meet any , , n or T R /3 F i x a , 0 , 0 , standard Each listed {(x,ix,t):x,t}, 2 T to: or T if T 2 R the o , , a Several assume T components o f F i x . in in may T T 4 p i s one 0 T /3 1 otherwise, T 4 p or 7 K 2 p - are involution Namely S XS X(1/2), 1 1-sphere, involutions standard under. . listed can consider S'xS'xO, in arise T=: more in the S'xixl, S'xS'xO, 11 3 {(x,x,t):x,t}, S x±ixl 1 It L and i ' same c a s e {(x,±ix,t);x,t}, tixS'xI, respectively. t o show involution if S'xtixl, 1 1 suffices standard the 1 and S x S x O , 1 that S xS xO, t has listed have the under isomorphic 1)- 3) t h e n they same fixed set as a t h e c o r r e s p o n d i n g c a s e , and fixed sets and fall into are conjugate. We make use o f Lemmas 9.1 a n d 9 . 2 , Corollary 7.4 and Corollary 8.4. Case 1) i T T = 0 n iT T decomposes u 3W^=T U i T . Since components k , k T 4 p t|(W,) for M into there or k is N I . Since with to k . Fixed T involutions C^T' isomorphic fixed between (hptBW^aWj >3W '=9W h:W, 2 a property does a sets T' o not i s conjugate fixed to s e t assume o f t h e same t y p e a s ^T4P r a r and obtained. e s e t , l e t h:W >W ' be a 2 2 i|(W '). Then 2 conjugation for i a n d t' boundary k which by t h e extends T to a are conjugate. separate and t does not the c o l l a r . M is >T x-1 involution is I 1 > W , ' . Hence 2) iT=T, T interchange d:T x1 2 extendable conjugation Then t|(W ) 2 conjugation s o t|(W^) t has a 2-dimensional conjugation with 2 i s a f i x e d s e t , tW^=W^. The conjugate t' Case two components W^ = T x l o f W^ a r e i n t e r c h a n g e d the standard Given a n d T does n o t s e p a r a t e . 2 isomorphic is an X on W t h a t to W/d isomorphism. where t is does n o t i n t e r c h a n g e W=T xI 2 induced the and by an components of 9W. Since X has a two dimensional f i x e d component, conjugate t o k X is or k « A 2 A Case 2 . 1 ) X i s conjugate t o k Then 9 F i x ( X ) = S , S A > u has 2 two components so Tr-|Fix has one component. O r i e n t S,. annulus Fix(X) induces an o r i e n t a t i o n on S . Then If d*[S,]=-[S J 2 2 then F i x i s a K l e i n b o t t l e meeting T. So d * [ S ] = [ S 3 and F i x 2 1 is a torus. into this i ' be conjugate to t and assume i ' Let case also. Construct Construct a conjugation T x1 >(T X1)' 2 Extend h(T x-1)=d'.h.(d 2 1 ) . Since extends over conjugation extendable conjugation h:W >W c o n j u g a t i o n between a bicollar p r o p e r t y of k between X A and components A, 2 A and 1-spheres. Let S ^ j = T x ( - 1 ) >9W by of the shows h extends to a X'. h induces 1 n and 2 Then Fix(X) 2) 1 A j . Pick o r i e n t a t i o n s two i n two so that >T x-1 2 2 or 2 1 S . 2 2 subcases: 2 2 1 T the same element of H,(W). d : T x 1 2 d(S, )=S a has F i x meets 1) d ( S ) = S , and d*[S,,] = [ S , ]. Then F i x i s two 1 1 defining F i x . Then i s o r i e n t a t i o n p r e s e r v i n g and must take S,, to S There are four and t and i ' 2 A represents above. F i x i s a t o r u s i n an o r i e n t a b l e Case 2 . 2 ) X i s conjugate to k « each as 2 to a c o n j u g a t i o n h:9W manifold, h annular = M' between X | ( T X 1 ) 2 2 X'|(T x1)'. W/6" falls and d*[S,,]=-[S , ] . 2 Then tori. F i x i s two K l e i n b o t t l e s . These meet T so t h i s case does not occur. 3) d ( S l 1 )=S 2 2 and d*[S,,]=[S ]. 2 2 Then Fix is a 1 15 nonseparating of torus since d must interchange the components W-Fix. 4) d(S 1 1 )=S separating If of is these also Case 3) is i then So a i n Case conjugate it to ]. Then Fix is one 3.1) B o t h X, and X Case 3.2) B o t h X, and X case Fix is two K l e i n just one K l e i n conjugation 3.1) fall conjugation into can the same one be constructed 2.1. Then M = W , W U k , k R 2 M is , w i t h T=W, 2 I-bundles i|(W^) 2 over a orientation k , k A A 2 P' k k 2 are conjugate to k , are conjugate to k by symmetry bottles, bottle. by t a k i n g t o W, will n W 2 Klein reversing. or 2 p k^. follows: Case In it twisted tW,=W, and ilaw^ilSWj extending 2 2 T separates. orientable We have So X = t | W Since as iT=T, where W\ a r e i given subcases. X and X' bottle. d*[S,,]=-[S torus. i' between and 2 2 If assume X, is a Klein bottle i' also is R 2 M , 2 p k A or k or k conjugate N A 2 p to and two p o i n t s , given construct any conjugation between X u s i n g the conjugation extendable 2 and X ' 2 property . k . R or a and of k . K In case 3.2) construction we X^ is not conjugate it to fixed set avoided K l e i n conjugate Next, is the is k to k . Let bottles intersects meeting T . T. By Therefore 2 M < possible A 2 p always that Fix n T X, is = S 1 conjugate U S 2 . Torus to k T A and X 2 induces an orientation k. the as T annular on S 2 f i x e d set does. In orientation on S would be a K l e i n the is set is 2 easy and k 2 p A is a v e t n e i e a n r >ular on S, is fixed. same o r i e n t a t i o n fixed set In on S induces an 2 the one T i n d u c e s . T h e r e f o r e T bottle. and X or a torus to c o n s t r u c t n ' opposite to a torus f i x e d set induces the t Hence we have X, fixed once an o r i e n t a t i o n X, are 2 conjugate and X 2 are conjugation k conjugate and f o u r p o i n t s . a conjugation to If between extendable i' i' and A to is and k^p the a n ^ given it i since k A property. QED 1 17 Bibliography [I] R. H. B i n g , A homeomorphi sm the sum of two solid horned between spheres, [2] M. Dehn, I dungsklassen, [3] A. C. [4] W. Hantzsche (1952), 354-362. Die 69 ( 1 9 3 8 ) , Gruppe G o r d o n and surfaces der 135-206. in R. A. H. Wendt, branched euklidische and Abbi Incompressible Drei Math. dimensi Ann. onale 110 J . H e m p e l , 3-manifolds, Princeton Univ. Press, [6] P. PL K. Kim and involutions [7] P. K. [8] K. Kwun and [9] G. R. [10] E. L u f t , and tori [II] E. L u f t and D. S j e r v e , Involutions with points on orientable flat 3-di me ns i o nal Kim and 3-manifolds, 221-237. Trans. the Amer. J. 29 J. Livesay, Soc. Involutions Ann. Equivariant with respect to appear in T r a n s . [12] P. Orlik, Seifert [13] C. R o u r k e and Pi ecewi Grenz. [14] J. se-Li folds, PL PL 203 of with near two 78 of 232 (1977), fixed sing , fibered 1 97-106 points (1963), 1 S'xS^S , of (1975), Math. Amer. M a t h . 1972. Sanderson, Topology, Verlag, Involutions Topology 20 Soc. Lecture manifolds, B. involutions S over involutions Soc. 86, on 582-593. surgery on incompressible annuli lo involutions, to a p p e a r . Verlag, 69, Springer Tollefson, 3-mani Studies 101-110. Amer. M a t h . Tollefson, Math. Springer (1982), Tollefson, Trans. three-sphere, 291, A n n . of M a t h . 1976 (1935), D. E. S a n d e r s o n , Orientation-rever on orientable torus bundles J., Math. preprint. [5] Michigan Math. Acta Litherland, coverings, Raumformen, 593-61 1 . the 3-sphere and A n n . of M a t h . 56 isolated space Notes Introduction Erg. of der 1972. sufficiently (1981), in fixed forms, Math. to Math. 323-352. u. ihrer large 118 [15] J . Wolf, 1967. [16] F. W a l d h a u s e n , Vber Involutionen Topology 8 (1969), 81-91. Spaces of constant curvature, der McGraw-Hill, 3-Sph'dre,
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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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Title | Involutions with 1- or 2-dimensional fixed point sets on orientable torus bundles over a 1-sphere and on unions of orientable twisted I-bundles over a Klein bottle |
Creator |
Holzmann, Wolfgang Herbert |
Publisher | University of British Columbia |
Date Issued | 1984 |
Description | We obtain a complete equivariant torus theorem for involutions on 3-manifolds M. M is not required to be orientable nor is H,(M) restricted to be infinite. The proof proceeds by a surgery argument. Similar theorems are given for annuli and for discs. These are used to classify involutions on various spaces such as orientable twisted I-bundles over a Klein bottle. Next we restrict our attention to orientable torus bundles over S¹ or unions of orientable twisted I-bundles over a Klein bottle. The equivariant torus theorem is applied to the problem of determining which of these spaces have involutions with 1-dimensional fixed point sets. It is shown that the fixed point set must be one, two, three, or four 1-spheres. Matrix conditions that determine which of these spaces have involutions with a given number of V-spheres as the fixed point sets are obtained. The involutions with 2-dimensional fixed point sets on orientable torus bundles over S¹ and on unions of orientable twisted I-bundles over a Klein bottle are classified. Only the orientable flat 3-space forms M₁, M₂ and M₆ have involutions with 2-dimensional fixed sets. Up to conjugacy, M₁ has two involutions, M₂ has four involutions, and M₆ has a unique involution. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-06-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080350 |
URI | http://hdl.handle.net/2429/25303 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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