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UBC Theses and Dissertations

Heegaard diagrams and applications Li, Zhongmou 2000

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H E E G A A R D D I A G R A M S A N D A P P L I C A T I O N S B y Z h o n g m o u L i M . Sc . ( M a t h e m a t i c s ) J i l i n U n i v e r s i t y A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES MATHEMATICS W e accept th is thesis as con fo rming to the requi red s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA M a y 2000 © Z h o n g m o u L i , 2000 In present ing th is thesis i n p a r t i a l fulf i lment of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree tha t the L i b r a r y sha l l make i t freely ava i lab le for reference a n d s tudy. I further agree tha t pe rmis s ion for extensive c o p y i n g of th is thesis for scho la r ly purposes m a y be granted by the head of m y depar tment or by his or her representat ives. It is unde r s tood tha t c o p y i n g or p u b l i c a t i o n o f th is thesis for f inanc ia l ga in sha l l not be a l lowed w i t h o u t m y w r i t t e n pe rmiss ion . M a t h e m a t i c s T h e U n i v e r s i t y of B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1Z1 D a t e : Abstract T h e m a i n objec t ive of th i s thesis is to s tudy Heegaard d i ag rams a n d the i r app l i ca t ions . F i r s t , we invest igate Heegaard d i ag rams of closed 3-manifolds a n d in t roduce the c i rc le a n d chord presenta t ion for a connected, closed 3-mani fo ld . T h e equivalence p r o b l e m for Heegaa rd d i ag rams after connected s u m moves a n d D e h n twis ts w i l l be inves t iga ted . Presen ta t ions w i l l be used to detect reduc ib le Heegaard d i ag rams a n d h o m e o m o r p h i c 3-manifo lds . W e also invest igate Heegaard d i ag rams of the 3-sphere. T h e m a i n resul t of th i s pa r t is t ha t i f two Heegaa rd d i ag rams of the 3-sphere have the same genus, t hen there is a sequence of connected s u m moves a n d D e h n twis t s to pass f rom one to the other . If we use connec ted s u m moves only , Heegaard curves can be changed to p r i m i t i v e curves a n d i f we use D e h n twists o n l y Heegaard curves can be brought in to a s imp le p o s i t i o n . F i n a l l y , we cons t ruc t an i m m e r s i o n of a compac t , or ientable , connected 3 -man i fo ld w i t h non -empty b o u n d a r y in to M 3 w i t h at mos t double a n d t r i p l e po in t s as s ingula r i t i es . Fu r the r , we prove tha t i f the b o u n d a r y of the 3 -man i fo ld consists of 2-spheres a n d the 3 -man i fo ld can immerse in to R 3 w i t h o n l y double po in t s as s ingular i t ies , then the 3-m a n i f o l d mus t be a p u n c t u r e d 3-sphere or a p u n c t u r e d ( S 1 x § 2 ) j j • • • (((S 1 x § 2 ) . i i Table of Contents Abstract N i i List of Figures v Acknowledgement v i i Introduction vi i i 1 Notation and Preliminaries 1 2 Geometry and Algebra of Heegaard splittings 9 2.1 C o n n e c t e d s u m move 9 2.2 D e h n twis t s 20 2.3 S t a b i l i z a t i o n a n d r educ t ion 22 2.4 A p rope r ty of moves 25 2.5 C i r c l e a n d cho rd presentat ions of closed 3-manifolds 26 2.6 D e t e c t i o n of c losed h o m e o m o r p h i c 3-manifolds 37 3 Heegaard diagrams of the 3-sphere S 3 41 3.1 S i m p l i f y i n g Heegaard d i ag rams of S 3 43 3.2 A n example 56 3.3 U s i n g D e h n twis t s o n Heegaard d i ag rams of S 3 57 4 Immersing orientable 3-manifolds into K 3 62 4.1 E v e n Heegaard d i ag rams 65 i i i 4.2 Proof of Theorem 4.1 72 4.3 Proof of Theorem 4.2 79 4.4 A n immersion of RP 0 3 , the punctured 3-dimensional projective space, into R 3 82 Bibliography 86 iv List of Figures 1.1 C o n n e c t e d s u m move 3 1.2 D e h n twis t 4 1.3 T w i s t hand le 4 1.4 H a n d l e a d d i t i o n 5 2.5 R e a d i n g w o r d of the in tersec t ion po in t s 10 2.6 A Heegaa rd d i a g r a m of the 3-sphere 18 2.7 H a n d l e a d d i t i o n move and connected s u m move 24 2.8 A c i rc le a n d chord presenta t ion for the Po inca r e h o m o l o g y 3-sphere . . . 27 2.9 C i r c l e a n d cho rd presentat ions for the lens spaces L(7, q), q = 1 , 2 , 3 , 4 , 5 , 6 33 2.10 A c i rc le a n d cho rd presenta t ion of S 3 35 3.11 S t a n d a r d handle i n R 3 42 3.12 S t a n d a r d Heegaa rd d i a g r a m of S 3 42 3.13 A h a n d l e b o d y w i t h two s t a n d a r d comple te systems i n M 3 44 3.14 C u t t i n g dV a long the comple te m e r i d i a n sys tem 45 3.15 U s i n g two D e h n twis t s to remove in tersec t ion po in t s 46 3.16 U s i n g D e h n twis t s to change curves to s t a n d a r d pos i t ions . 52 3.17 T w o s t rong ly equivalent Heegaard d i ag rams of ( S 1 x § 2 ) { j R P 3 56 3.18 A Heegaard d i a g r a m of S 3 57 3.19 U s i n g a connected s u m move to s imp l i fy the Heegaard d i a g r a m 58 3.20 U s i n g D e h n twis t s to s imp l i fy the Heegaard d i a g r a m 59 v 4.21 A Heegaard d i a g r a m of S 3 tha t is not even 66 4.22 Curves i n dV t ha t separate dV as d isks 67 4.23 G e t t i n g a d i a g r a m for Heegaard a n n u l i 68 4.24 D e h n twis t a long D 71 4.25 C h a n g i n g crossings i n a d i a g r a m of a n n u l i 73 4.26 Re idemei s t e r moves 74 4.27 C h a n g i n g crossings i n a d i a g r a m of a n n u l i 77 4.28 A d d i n g a 3-cel l to change cross ing 78 4.29 A Heegaard d i a g r a m of I P 3 82 4.30 A p ro j ec t i on f rom R + to the x y - p l a n e 84 4.31 A n i m m e r s i o n of the s o l i d torus in to R 3 85 v i Acknowledgement I w o u l d l ike to t h a n k m y advisor , D r . D a l e Ro l f sen , who spent m u c h t i m e to guide me t h r o u g h m y research a n d p r o v i d e d inva luab le advice . I w o u l d also l ike to t h a n k D r . E r h a r d L u f t for r ead ing t h r o u g h an ear l ier draft of th i s thesis, his comments , a n d his s ignif icant suggestions. M a n y t hanks to D r . D a v i d G i l l m a n w h o has h a d signif icant influence on the progress o f m y s tudies . v i i Introduction In the ea r ly 1960s, W . H a k e n [13], [12] i n t roduced the theory o f n o r m a l surfaces a n d he e x h i b i t e d an a l g o r i t h m to detect embedded , 2-sided, c losed, incompress ib le surfaces i n c losed, i r r educ ib le 3-manifolds . H i s a l g o r i t h m w i l l eventua l ly s top i f such surfaces exis t . B u t the a l g o r i t h m h a d no b o u n d for t e r m i n a t i o n . W . J a c o a n d U . O e r t e l [19] ob t a ined an efficient a l g o r i t h m w i t h such a b o u n d . Jaco a n d Oer te l ' s a l g o r i t h m a n d a resul t o n the conjugacy p r o b l e m i n the m a p p i n g class group of surfaces ( H e m i o n [16] or H a t c h e r a n d T h u r s t o n [14]) gives an a l g o r i t h m to decide i f two closed, i r r educ ib le 3-mani fo lds are h o m e o m o r p h i c a s suming one of the 3-manifolds conta ins an embedded 2-sided incompress ib le surface. J . H . R u b i n s t e i n [33], [34] genera l ized n o r m a l surfaces to a lmos t n o r m a l surfaces and used t h e m to solve the r ecogn i t ion p r o b l e m for the 3-sphere, i.e., to decide i f a g iven closed 3 -man i fo ld is the 3-sphere. A . T h o m p s o n [38] also gave an a l g o r i t h m to recognize the 3-sphere by us ing t h i n p o s i t i o n . In the first pa r t of the thesis, we invest igate Heegaard d i ag rams of c losed 3-manifo lds a n d in t roduce a new representa t ion m e t h o d for connected, c losed 3-manifolds . R e c a l l t ha t a Heegaa rd s p l i t t i n g of genus g of a closed 3 -man i fo ld M decomposes M i n to two h a n -d lebodies V , W of genus g. T h e b o u n d a r y curves bi, b2, • • •, bg of the d isks i n a comple te m e r i d i a n d i sk sys t em i n W (i.e., c u t t i n g W a long the disks is a 3-cell) l ie i n the surface dV as a comple te sys t em (i.e., c u t t i n g dV a long the curves is a d i sk ) . T h e (one-sided) Heegaa rd d i a g r a m ( V ; &i, b2, • • •, bg) comple t e ly determines M. A f t e r choos ing a su i tab le comple te m e r i d i a n d i sk sys tem D\, D2, • • • , D s of V , we can find sub-arcs Ci, c 2 , • • •, c 2 s _i of the curves b\, b2, • • •, bg whose end-points l ie i n the curves dD\, dD2, • • •, dDg such tha t c u t t i n g dV a long the sub-arcs ci, c 2 , • • • , c2g-\ a n d the curves dDi, dD2, • • • , 8Dg leaves v i i i a d isk . T h e r e m a i n i n g par t s of the Heegaard curves bi,b2,---,bg are chords i n the disk . W e can use the d isk , the chords a n d the labels w h i c h denote the in te rsec t ion po in t s be-tween the curve set {bx,b2,---,bg} a n d the curve set {dDi,dD2,•••,dDg} to represent the 3 -man i fo ld M. W e give a necessary a n d sufficient c o n d i t i o n tha t a c i rc le w i t h some chords a n d l abe led endpoin t s represents a 3 -man i fo ld ( T h e o r e m 2.6). W e also invest igate the equivalence p r o b l e m for Heegaard sp l i t t i ngs o f c losed 3-man i fo lds . W e prove tha t a Heegaard d i a g r a m ( V ; 61, b2, • • •, bg) c an be changed to an-other Heegaa rd d i a g r a m (V; b'^b^,---, b'g) by us ing a sequence of connected s u m moves i f a n d o n l y i f b[, b'2, • • •, b'g have t r i v i a l reduced words co r respond ing to the comple te sys tem bi,b2, - • • ,bg of the surface dV ( T h e o r e m 2.2). P rope r t i e s of connected s u m moves a n d D e h n twis t s are inves t iga ted . C o m b i n i n g these results w i t h results abou t s t a b i l i z a t i o n of Heegaa rd sp l i t t i ngs ([11], [36], [23], [20], [35]), we o b t a i n an a l g o r i t h m to detect i f two closed, connec ted 3-manifolds are h o m e o m o r p h i c ( T h e o r e m 2.12). It is an open p r o b l e m to f ind a b o u n d for th is a l g o r i t h m . In the second par t of the thesis, Heegaard d i ag rams of the 3-sphere are inves t iga ted . T h e m a i n resul t of th i s pa r t is tha t i f two Heegaard d i ag rams of the 3-sphere have the same genus, t hen there is a sequence of connected s u m moves a n d D e h n twis t s to pass f r o m one to the o ther ( T h e o r e m 3.3). F o r every Heegaard d i a g r a m (V; bi, b2, • • •, bg) of the 3-sphere, we prove the fo l lowing two proper t ies . F i r s t , i f the h a n d l e b o d y V l ies i n the 3 -d imens iona l E u c l i d e a n space R 3 s t a n d a r d l y (i.e., R 3 - Int V is h o m e o m o r p h i c to a h a n d l e b o d y of genus g w i t h a 3-cell removed) , t hen we can use D e h n twis t s on V to change the Heegaa rd curves h,b2, - • • ,bg to new Heegaard curves b[,b'2, • • •,b'g such tha t b'ijb^, • • • ,b'g b o u n d pa i rwise d is jo in t disks i n R 3 — Int V ( T h e o r e m 3.5). Second , we can use connec ted s u m moves on the curves bi, b2, • • •, bg to o b t a i n new Heegaard curves bl1,b2,-"i b'g such tha t b[, b'2, • • •, b' are p r i m i t i v e curves of V; i.e., they f o r m a free base of the fundamen ta l g roup of the h a n d l e b o d y V ( T h e o r e m 3.4). i x In the t h i r d pa r t o f the thesis, we cons t ruc t an i m m e r s i o n of any g iven compac t , or ientable , connected 3 -man i fo ld w i t h non-empty b o u n d a r y in to R 3 w i t h at mos t double a n d t r i p l e po in t s as s ingular i t i es by us ing a Heegaard d i a g r a m of the 3 -man i fo ld (Theo-r e m 4.1). T h i s is s t ronger vers ion of a spec ia l case of a result o f J . H . C . W h i t e h e a d [41] w h i c h does not have a b o u n d on the m u l t i p l i c i t y o f the s ingular i t i es . Fu r the r , we prove t ha t i f a compac t , or ientable , connected p u n c t u r e d 3 -mani fo ld can immerse in to R 3 w i t h o n l y doub le po in t s as s ingular i t ies , then the 3 -mani fo ld must be a p u n c t u r e d 3-sphere or a p u n c t u r e d {S1 x S2)^1 x S2)$ • -^{Sl x S2) ( T h e o r e m 4.2). Chapter 1 Notation and Preliminaries W e w i l l work t h roughou t i n the piecewise l inear category. A l l mani fo lds a n d a l l maps are piecewise l inear . A l l in te rsec t ion po in t s are i n general p o s i t i o n . T h e t e r m homeo-m o r p h i s m always means a piecewise l inear h o m e o m o r p h i s m . O u r reference is [17] a n d [31]. A 3 -man i fo ld V is ca l l ed a handlebody of genus g i f there exis t g d is jo in t p roper 2-cells Di, • • •, Dg i n V such tha t i f Di x [—e, e],i = 1, • • • , g are d is jo in t regular ne ighborhoods of Di = Di x 0, i = 1, • • • , g i n V t hen C = V - {Dx x ( - e , e) U • • • U Dg x ( - e , e)) is a 3-cel l . T h e d isks Di, - • • ,Dg are ca l l ed a complete meridian system of V. T h e 3-cel l C is ca l l ed a cut o f V a long Di, • • •, Dg. A Heegaard splitting (V, W) of genus g of the closed connected 3 -man i fo ld M consists o f two hand lebod ies V, W of genus g i n M such tha t M = VuW a n d VC\W = dV = dW. If {Bi, • • •, Bg} is a comple te m e r i d i a n d i sk sys tem of W a n d dBi = bi, i = 1, • • • , g, t hen (V; bi, • • •, bg) is ca l l ed a (one-sided) Heegaard diagram of genus g o f M. N o t e t ha t a Heegaa rd d i a g r a m determines the 3 -man i fo ld M (i.e., M c an be o b t a i n e d by a t t a c h i n g 2- handles a long dis jo in t regular ne ighborhoods bi x [—e, e], i = 1, • • • , g i n dV, p lus one 3- hand le . T w o Heegaa rd sp l i t t i ngs (V, W), ( V , W) of M are ca l l ed strongly equivalent i f there exists a n ambien t i so topy ht : M —> M, 0 < t < 1, such tha t h0 = identity a n d ^ ( V ; W) = {V, W). F o l l o w i n g B i r m a n [2], we c a l l two Heegaard sp l i t t ings (V, W), ( V , W) equivalent i f there exis t a h o m e o m o r p h i s m h : M —> M w i t h h(V) = V, h(W) = W 1 Chapter 1. Notation and Preliminaries 2 or h(V) = W, h(W) = V. Let (V, W) be a Heegaard splitting of genus g of the closed connected 3-manifold M. Let 7 C V be an arc and D C V be a disk such that 3D = 7U (DndV), #7 = jf)dV. If N(j) is a closed regular neighborhood of 7, then V = VUN(j) and W = C7(W-iVT/y)) are handlebodies of genus # + 1. Therefore, ( V , W ) is a Heegaard splitting of genus g + 1 of M. This construction from the Heegaard splitting (V, W) to ( V , W) is called handle addition. The inverse procedure is called a reduction. A Heegaard splitting is called minimal or irreducible if it can not be reduced. Reidemeister-Singer Theorem. Let M be a connected, closed 3-manifold and let (Vi,Wi), (V2, W2) be two Heegaard splittings of M. Then there exist Heegaard splitting (Vi,Wi),(V2,W2) of M such that (Vi,Wi) reduces to (VuWi) and (V2,W2) reduces to (V2,W2) and ( V i , ^ ) , (V2,W2) are strongly equivalent. Now, we state the definitions of three well-known fundamental moves on some curves in the boundary surface of a handlebody here. Definition 1.1 Suppose that V is a handlebody of genus g and C\, c2,---,cm are pairwise disjoint, simple closed curves in dV. Then we can use the following three kinds of moves to change the positions of the curves in dV. I. Connected sum or handle sliding. We replace the curve Ci by c ^ Q C j in the curve set C\,c2, - • • ,cm as in the following figure (where i,j = 1,2 ••• ,m;i ^ j, and a is an arc in dV whose endpoints P, Q lie in the curves C{ and Cj respectively such that a n ^ C k ) = {P,Q}. (See Fig. 1.1.) Note. Connected sum moves are defined in an orientable surface, i.e., we can ignore the handlebody itself and only consider the moves in its boundary. Note. If (V, W) is a Heegaard splitting of M and (V; bi, b2, • • •, bg) a corresponding Heegaard diagram defined by a complete meridian system B\, B2, • • •, Bg ofW with dBi = Chapter 1. Notation and Preliminaries 3 F i g u r e 1.1: C o n n e c t e d s u m move bi, i = 1,2, • • •, g, and ifb^cbj is a connected sum move (to replace bi) on dV = dW, then there is a complete meridian system P>i, • • •, J B J _ I , B, Bi+X, • • •, Bg ofW with dB = b4cbj. II. Dehn twist. Suppose D is a meridian disk in V, i.e., cutting V along D is con-nected. Then we use the following Dehn twist on the curve set. (See Fig. 1.2.) Note. If (V; h,b2, - • • ,bg) is a Heegaard diagram of M and b[,---, b'g are the curves obtained by applying a Dehn twist TD (where D is a properly embeded, non-separating disk in V) to h,b2, - • • ,bg. Then {V;^,^, - • • ,b'g) is a Heegaard diagram of M' with Heegaard splitting (V,W) such that there is a homeomorphism h : M —> M' with h(V, W) = (V, W) and h \ V = T D . Note. Sometimes we need to use one kind of move similar to a Dehn twist. Suppose D is a proper disk in V that separates V into two connected components. We rotate one of the connected components with angle n along the disk D. (See Fig. 1.3) III. Handle addition (or stabilization) and reduction. Suppose DX,D2 are two small disjoint disks in dV and (Dx U D2) n (U^Ci) = 0. Then we add a 1-handle to V along Chapter 1. Notation and Preliminaries F i g u r e 1.3: T w i s t hand le Chapter 1. Notation and Preliminaries 5 F i g u r e 1.4: H a n d l e a d d i t i o n Di, D2 and let cm+\ be a simple closed curve in the boundary of the 1-handle as in Fig. 1.4-That is, c m + i is the union of an arc in dV - (Int(Di)Ulnt(D2) U ( U ^ C j ) ) and an arc in the handle. The inverse move of a stabilization is called a reduction. A Heegaard splitting is called minimal or irreducible if it cannot be reduced, that is, it can not be obtained by a stabilization of a Heegaard splitting of smaller genus. Remark. It is clear tha t the inverse move o f a connected s u m or a D e h n twis t is s t i l l a connec ted s u m or a D e h n twis t , respect ively. A n d the inverse move of a hand le a d d i t i o n is a handle r educ t ion , i.e. r e m o v i n g a 1-handle f rom H a n d r e m o v i n g the curve cm i f there exists a p rope r embedded d isk D i n the 1-handle a n d a curve, say c m , i n the curve set such t ha t dD n ( U ^ c * ) = 3D n cm a n d 3D a n d cm intersect t ransverse ly at one po in t . Somet imes , we w i l l consider o ther spec ia l curves i n the b o u n d a r y o f a h a n d l e b o d y V. A s imp le c losed curve i n dV is ca l l ed primitive i f there is a m e r i d i a n curve i n dV such tha t the two curves intersect t ransversely at one po in t . A complete primitive system of V. Chapter 1. Notation and Preliminaries 6 is a complete system b2, • • •, bg} of dV which in addition satisfies the property that there exists a complete meridian system {d\, d 2 , • • •, dg} of V such that bi intersects di transversely at one point and bi does not intersect dj, where i, j = 1,2, • • •, g; i ^ j. The concept primitive refers to primitive elements of free groups. (See Definition 1.2.) Definition 1.2 A set of primitive elements of a free group F is a set of elements which can be completed to a set of free generators for F. Although primitive curves are curves which have a simple property, they may have infinite possible positions in the surface. By a result of Zieschang [43], also [10], if we choose a complete meridian system for the handlebody, then the word of a primitive curve corresponding to the complete system is conjugate to a word which can be extended to a basis of the fundamental group of the handlebody. It is clear that the above condition is a necessary and sufficient condition. By Whitehead's results about automorphisms of free groups [42], there is an algorithm to determine if an element w of a free group F of finite rank is or is not primitive. Therefore, this gives an algorithm to determine whether a curve in the boundary of a handlebody is a primitive curve or not. Gordon [10] proved another necessary and sufficient condition: A curve in the boundary of a handlebody of genus g is primitive if and only if adding a 2-handle to the handlebody along the curve yields a handlebody with genus g — 1. The above results of Zieschang and Gordon imply that to check whether a simple closed curve in the boundary of a handlebody is primitive or not it is only necessary to check whether or not the word of the curve with respect to a complete meridian system of the handlebody is conjugate to a primitive element in the free group generated by the elements which are corresponding to the curves in the complete meridian system. There is a well-known simple method to check whether a simple closed curve bounds a disk in a handlebody or not. Chapter 1. Notation and Preliminaries 7 Lemma 1.1 Supposse V is a handlebody of genus g and dx, d2, • • •, dg is a complete meridian system ofV. If D is a properly embedded disk in V then the reduced word of dD corresponding to the complete meridian system is 1, i.e., the empty word. On the other hand, suppose c is a simple closed curve on dV. If the cyclically reduced word of c corresponding to a complete meridian system is trivial, then c bounds a disk in the handlebody. Proof. S ince the fundamen ta l group o f V is a free g roup generated by the generators co r r e spond ing to the comple te m e r i d i a n sys tem a n d dD c an sh r ink to one p o i n t i n V, t hen [dD] representes the t r i v i a l element 1 i n the free group . Therefore , the reduced w o r d o f 3D co r r e spond ing to the comple te sys tem is 1. O n the o ther h a n d , i f the reduced w o r d o f c co r re spond ing t o a comple te m e r i d i a n sys tem is 1, t hen we consider a regular ne ighbo rhood TV o f the curve c i n the surface dV. Suppose i : N —>• V is the i n c l u s i o n m a p . Since ker(i* : ni(N) —> TTI(V)) ^ 0, t hen by the l o o p theo rem [29], there exists a p rope r ly embedded d isk D i n V such t ha t dD C N a n d dD c an not be moved to a po in t i n TV cont inuous ly . N o t e t ha t dD is a s imp le c losed curve. T h u s we m a y assume tha t the curve dD is jus t the curve c. Therefore , c bounds a d i sk D i n the hand lebody . • T h e r e are some fundamenta l results o n free groups w h i c h are useful for d e t e r m i n i n g p r i m i t i v e elements of a free group . Definition 1.3 (Nielsen transformation). Suppose that Wi,w2, • • - ,wn are words in the generator set X = {xx, x2, • • •, xn} of the free group F(X). An elementary Nielsen trans-formation of the word set W = {wi, w2, • • •, wn} is of one of the following three types: for some i, 1 < i < n, (1) replace x\ byxV1, Chapter 1. Notation and Preliminaries 8 (2) interchange xx and X{, leaving others the same, (3) replace x \ with X\x2, all others fixed. Nielsen Theorem. Suppose u2, • • • ,un} a n d {vx, v2, • • •, vn} are two sets o f words i n the generator set X = {xi,x2, • • • , xn} of a free g roup F(X). I f u2,---, un} a n d {vi, v2, • • •, vn} are bases of F(X), t hen a finite sequence o f e lementary N ie l s en t r ans fo rmat ions w i l l change « i , u2, • • •, un to vx, v2, • • •, v n . O n the other h a n d , B i r m a n es tabl ished a s imp le c r i t e r ion to show whether or not a w o r d set i s a basis o f a free goup F i n t e rms o f the free dif ferent ia l ca l cu lus w h i c h was i n t r o d u c e d by F o x i n [Fox]. Birman Theorem. A set of words {wi, w2, • • •, wn} i n the generator set X = {xi,x2, • • •, xn} of the free g roup F(X) is a basis for F(X) i f a n d o n l y i f F o x m a t r i x dvH [dxj)nxn is i nve r t ib le i n the group r i n g Z[F(X)\, where the F o x der iva t ive d dxj Z[F{X)] -> Z[F(X)] is g iven by 0 r i f T«l . R £ 2 T e r ) _ S r ^ f : . X , r e i e i _ i 2^ei-1). "3 i=l (where e; = 1 or — 1; a n d 8^ is the K r o n e c k e r 8). B i r m a n ' s t heo rem determines an a l g o r i t h m to see whether a curve o n the b o u n d a r y surface o f a h a n d l e b o d y is p r i m i t i v e or not . Chapter 2 Geometry and Algebra of Heegaard splittings In this chapter, M will always denote a closed, connected, orientable 3-manifold. In this chapter, we consider two operations on Heegaard diagrams of a 3-manifold: Connected sum move ( or called handle sliding by some authors ) and Dehn twist. We also introduce a new tool, the circle and chord presentation of a closed, orientable 3-manifolds. Using this presentation, we will obtain the following results. 1. An algorithm to list all possible circle and chord presentations of closed, orientable 3-manifolds. Furthermore, we obtain a method to list all closed orientable 3-manifolds. 2. An algorithm to detect whether the corresponding Heegaard diagram of a circle and chord presentation is reducible or not. This will also give a method to detect whether or not a closed, orientable 3-manifold is the 3-dimensional sphere. 3. Relations between strong equivalence and equivalence. 2.1 Connected sum move Definition 2.1 Let F be an orientable closed surface of genus g. A set of k pairwise disjoint, oriented, simple closed curves in F is called non-separating if the result of cutting F along these curves is connected. If furthermore, k = g, then we call the curve set a complete system of F. IfV is a handlebody of genus g, then an oriented curve c is called a meridian curve of V if c is a non-separating curve on dV and c bounds a proper disk D in V. D is called a meridian disk ofV. A complete meridian system ofV is a complete system of dV which in addition satisfies the property that every curve in the system is a 9 Chapter 2. Geometry and Algebra of Heegaard splittings 10 o The orientation of F The orientation of F c b: c 1). Reading off the intersection point as " b ; - i 2). Reading off the intersection point as " bj " F i g u r e 2.5: R e a d i n g w o r d o f the in tersec t ion po in t s meridian curve ofV. Definition 2.2 Suppose {b\, b2, • • •, b9} is a complete system of an oriented closed sur-face F of genus g. Suppose c is an oriented simple closed curve in F. If we travel along c in direction of its orientation starting at a point on c, and reading off the intersection points between c and the curve set {bi, b2, • • •, bg} according to the orientations of the respective two subarcs near each intersection point ( see Fig. 2.5 ), then we obtain a word w(bi,b2,- • • ,bg). We call the word w(bi, b2, • • •,bg) the word of c corresponding to the complete system {bi,b2,---,bg}. Note. In D e f i n i t i o n 2.2, since {&i , b2, • • •, bg} is a comple te sys tem of F , there is another comple te sys tem { a i , a2, • • •, ag} such tha t bi PI aj = 0 for i, j = 1,2, • • • , g; i ^ j a n d bi intersects at one po in t . A f t e r i so top ica l ly m o v i n g the curves & i , • • • , bg, ai, • • •, ag i n F to let t h e m be a t t ached to one po in t P G F, c u t t i n g F a long the new curves w h i c h are s t i l l denoted bx, • • •,bg, a i , • • •,ag leaves an open disk . Therefore , the fun-d a m e n t a l g roup iti(F) has 2g generators bi, b2, • • •, bg, a i , a2, • • •, ag a n d a re la tor R = Chapter 2. Geometry and Algebra of Heegaard splittings 11 axbiai~1bi~1a2b2a2~1b2~1 • • • a 5 6 s a s _ 1 6 i _ 1 . B y the Independence T h e o r e m ( see T h e o -r e m 4.10 i n [26] ), the subgroup of 7Ti(F) generated by bx, 6 2 , • • •, b9 is a free g roup a n d w(bi, • • •, b9) is an element of th is free group. Note. W h e n we i s o t o p i c a l l y move the curve c to a new curve d i n F, the w o r d w' o f d co r r e spond ing to the comple te sys tem is g iven by inse r t ing or r e m o v i n g some cance l l i ng pa i rs of generators. A m o n g the equivalent moves of Heegaard d iagrams , connected sum moves ( or handle sliding ) is the mos t i m p o r t a n t of the moves because of the fo l l owing two reasons. O n e is t ha t o n l y connected sum moves c an decrease lengths of c y c l i c a l l y reduced words o f Heegaard curves. A n o t h e r reason is tha t o n l y connected sum moves c an be used to o b t a i n new p r i m i t i v e curves w h i c h can be used to reduce Heegaard d i a g r a m . W e general ize a resul t i n [37] to o b t a i n a n a lgebra ic p rope r ty o f the connec ted s u m move. Theorem 2.1 Suppose F is an oriented surface of genus g and curve set {hi , 62, • • •, bg} is a complete system in F. Suppose a\ is a simple closed curve in F and a\ does not sep-arate F. If the reduced word w of ai corresponding to the complete system {b\, 6 2 , • • •, bg} is 1, i.e., empty word, then we can use a sequence of connected sum moves on the curve set {bi, b2, • • •, bg} to obtain a new complete system { a 1 ; a 2 , • • •, ag}. In particular, if the length of w is s, then at most (s + l)(g — 1) moves are required.. In the p r o o f of T h e o r e m 2.1, we w i l l make use of the fo l lowing two l emmas . Lemma 2.1 Suppose F, {bi, b2, • • •, bg} and a\ are the same as in Theorem 2.1. Suppose that we use a connected sum move on the curves bx,b2 along an arc a to replace bi by b[ — bi^ab2 to obtain a new complete system {b[, b2, • • •, bg}. Suppose that the words of a\ corresponding to the old complete system {bx, b2, • • •, bg} and the new complete system Chapter 2. Geometry and Algebra of Heegaard splittings 12 {b[, b2, • • •, b9} are w{b\,b2, • • •, bg) and, if/(fr'i, b2, • • •, bg) respectively. If the reduced word of w(b±, b2, • • •, bg) is 1, then the reduced word of w'(b[, b2, • • •, bg) is also 1. Proof. W i t h o u t loss o f general i ty, we m a y suppose t ha t the o r i en t a t i on o f b[ matches the or ien ta t ions of b o t h b\ a n d b2. In an in tersec t ion po in t between a a n d a\, we read n o t h i n g for the o l d comple te sys tem and read a cance l l i ng pa i r b[ • (b[)~l or • b[ for the new comple te sys tem. In an in tersec t ion po in t between 61 a n d ax, i f we read 61 or for the o l d comple te sys tem, t hen we read b[ or respect ive ly for the new comple t e sys tem. I n an in te rsec t ion p o i n t between b2 a n d a i , i f we r e a d b2 or (b2)~l for the o l d comple te sys tem, then we read b[ • b2 or (b[ • b2)_1 respect ive ly for the new comple te sys tem. In a l l other in te rsec t ion poin ts , we read the same results for b o t h the comple te systems. Therefore , w'(b[, b2, • • •, bg) has the same reduced w o r d w i t h a w o r d w(b[, b[ • b2, • • •, bg). T h e reduced w o r d of w^'^b^ • b2, • • •, bg) is 1 since the reduced w o r d of w(bi, b2, • • •, bg) is 1. T h u s , the reduced w o r d o f w'(b[, b2,---,bg) is also 1. • Lemma 2.2 Suppose F, {bx, b2, • • •, bg} and ax are the same as in Theorem 2.1. If a\ D (U9j=1bj) = 0, then we can use at most g — 1 connected sum moves to change the complete system {61, b2, • • •, bg} to a new complete system {ax, a2, • • •, ag}. Proof. T h e resul t of c u t t i n g F a long the curves 61, b2, • • •, bg is a 2-sphere w i t h 2g holes. Deno te th is surface as E a n d the holes as b\, b\, b\, b\, • • •, bg, b2g, where 6], b2 are two copies of bj for j = 1,2, • • • , g. I f we cut E a long a i , we o b t a i n two surfaces E i , E 2 w h i c h are 2-spheres w i t h holes. O n e of the surface E i , E 2 has at mos t g + 1 b o u n d a r y connected components . W i t h o u t loss of general i ty, we assume the surface is E i . Since ax does not separate F, there is some i 6 { 1 , 2 , • • •, g} such tha t b] C E i , b\ C E 2 or b\ C E2, b\ C E i . W i t h o u t loss of general i ty, we m a y assume tha t b\ C E i , bf C E 2 - Suppose Chapter 2. Geometry and Algebra of Heegaard splittings 13 t ha t the n u m b e r of the holes of E x is k. W e d raw k - 2 p r o p e r l y embedded , pa i rwise d is jo in t arcs i n E i such tha t each hole of E i except b\ a n d a\ is connec ted to the hole b\ by e x a c t l y one arc. N o w , these arcs define k — 2 connected s u m moves. T h e resul t curve of the moves is i so top ic to a i i n the surface F. Therefore , we o b t a i n a new comple te sys tem { a i , a2, • • •, a9}. I n fact, the curves a2, • • •, a9 are &i , • • • , tV-i , • • • , b9. T h e n u m b e r of moves is k — 2 < g — 1 since k < g + 1. • Proof of Theorem 2.1. I f a i n (Lr? = 16j) ^ 0, we read the w o r d w of a\ coo respond ing to the comple te sys tem {bi, b2, • • •, b9}. Since the c y c l i c a l l y reduced w o r d of w is 1, the e m p t y w o r d , we can reduce w to 1 by cance l l i ng reduced pa i rs of w step by step. Suppose the first such reduced pa i r i n w is bi a n d bT1 for some i e { 1 , 2 , • • •, g}. T h e n bi a n d bT1 are adjacent i n the presenta t ion o f w. T h i s fact indica tes tha t there exists a n arc c C fli so t ha t c intersects ( U ^ f y ) by two po in t s A a n d B w h i c h are the endpoin t s of c. T h e po in t s A, B separate the curve bi i n to two arcs a a n d b. T h e resul t o f c u t t i n g F a long the curves bi,b2,---,bg is a 2-sphere w i t h 2g holes a n d we denote th i s surface as E . If we cu t the surface E a long c, we o b t a i n two surfaces Ex a n d E 2 . O n e of the surfaces E i a n d E 2 does not i nc lude the hole w h i c h is the copy of bi. W e assume, w i t h o u t loss of general i ty, th i s surface is E i and assume tha t E i inc ludes the hole c U a. N o w , we v i e w E x as a d i sk b o u n d e d by c U a a n d w i t h k holes ins ide . W e d raw A; p r o p e r l y embedded , pa i rwise d is jo in t arcs i n E x such tha t each o f the holes is connected to the curve a by exac t ly one arc. N o w , we go back to consider the surface F. A l o n g each of the arcs, we use the respect ive connected s u m move. These moves do not remove the curves & i , • • • , & t _ i , frj+ii • • • i bg f rom the new comple te sys tem, i.e., we replace the curve bi i n the comple te sys tem {h,b2, - • • ,b9} w i t h the curve w h i c h is the connected sums o f bi w i t h the k curves i n the curve set {bx, • • • , 6 j + i , • • • , b9} a long the k arcs. N o t e tha t th i s curve (up to i so topy i n E ) is jus t c U b. T h e n , we o b t a i n a new comple te sys t em Chapter 2. Geometry and Algebra of Heegaard splittings 14 {61, • • • , c U b,bi+i, • • • ,b9} w h i c h has at least two less in te rsec t ion po in t s w i t h the curve di a n d the reduced w o r d of ax co r respond ing to the new comple te sys tem is s t i l l 1 by L e m m a 2.1. N o t e tha t the number of our moves is k w h i c h is at mos t 2g - 2. Fu r the r , we reduced one cance l l i ng pa i r bi a n d bT1 i n the representa t ion of w. W e cont inue the above step to the adjacent cance l l i ng pa i r of the w o r d w a n d finally we w i l l get a comple te sys t em {b[, b'2, • • •, b'g} o f the surface F so t h a t every curve i n th i s comple te sys tem does not intersect the curve ax. T h e number of the moves is at mos t s/2 x (2#-2) . N o w , ax n (Uf=1&9 = 0. Therefore , by L e m m a 2.2, we can use at mos t g — 1 connec ted s u m moves on the comple te sys tem {b'x, b'2, • • •, b'g} to o b t a i n a new comple te sys tem {ax,a2, - • • ,ag}. Since we o n l y need s/2 x (2g — 2) t imes of connected s u m moves to cancel the s in te r sec t ion po in t s between ax a n d the comple te sys tem bx,b2, - • • ,bg a n d need at mos t g — 1 more moves to get the curve ax, the number of t o t a l connected s u m moves is at mos t s/2 x (2g — 2) + (g — 1) = (s + l)(g — 1). T h i s completes the p r o o f o f the theo rem. • N o w , we a p p l y T h e o r e m 2.1 to get several results . Theorem 2.2 Suppose F is an orientable surface of genus g and {h,b2, • • • ,bg} is a complete system in F. Suppose {ax, a2, • • •,an}(n < g) is a non-separating curve set in F, i.e., cutting F along {a\,a2, • • • ,an} is connected. If the respective reduced words Wy,w2, • • • ,wn of ai, a2, • • •, an corresponding to the complete system are all equal to 1, then we can use a sequence of connected sum moves on the complete system to obtain a new complete system {ax, a2, • • •, an, b'n+1, • • •, b'g}. If the sum of the word lengths of wx,w2,---,wn is s, then at most (s + n)(g — 1) moves are required. In particular, if n = g, then the requirement that all Wi reduce to 1 gives a necessary and sufficient Chapter 2. Geometry and Algebra of Heegaard splittings 15 condition passing one Heegaard diagram to another Heegaard diagram by using connected sum moves. Proof. T h e p r o o f is s i m i l a r to T h e o r e m 2.1. W e cancel a l l in te rsec t ion po in t s be-tween the curve set {&i, b2, • • •, bg} a n d {ai, a 2 , • • •, an} first, t ha t is, we can use at mos t s(g — 1) moves o n the comple te sys tem {bi, 6 2 , • • •, bg} to o b t a i n a new comple te sys tem {b'[, b2, • • •, bg} such t ha t bi, b2, • • •, bg, b", b2, • • •, b"g are pa i rwise d is jo in t curves i n F. W e cut F a long {&", b2, • • •, b"g} to o b t a i n a 2-sphere w i t h 2g holes. Deno te th i s surface as E . T h e r e is a curve, say a x w i t h o u t loss o f generali ty, i n the curve set {ai, a 2 , • • •, an} such tha t ax bounds a d isk w i t h m holes i n E w i t h m < g and no curves i n the curve set {ai ,a 2 , • • •,an} l ie ins ide th is p u n c t u r e d disk . Deno te th is p u n c t u r e d d i sk as E i . B y the p r o o f of T h e o r e m 2.1, we can use at most g — 1 connected s u m moves to o b t a i n a\. W e cont inue the s i m i l a r steps for other curves i n the curve set {ai, a 2 , • • • , an} a n d finally get a comple t e sy s t em {ai, a 2 , • • • , a „ , b'n+1, • • •, b'g}. T h e n u m b e r of t o t a l moves is at mos t s{g - 1) + n(g - 1) = (s + n)(g-l). • Theorem 2.3 Suppose that (V; b\, b2, • • •, bg) is a Heegaard diagram of genus g of M. If one of the following conditions is true, then the Heegaard diagram is reducible ( i.e., we can use a reduction move on the Heegaard diagram to obtain a Heegaard diagram of genus g — 1 ) . 1) . One of the Heegaard curves b\,b2,---,bg is primitive in the handlebody V. 2) . There is a simple closed curve c in dV such that c is primitive in V and c can be obtained by using a sequence of connected sum moves on the curve set b\, b2, • • •, bg. 3) . There is a simple closed curve c in dV such that c is primitive in V and the reduced word of the curve c corresponding to the complete system b2, • • •, bg} of dV is 1. Chapter 2. Geometry and Algebra of Heegaard splittings 16 Proof. 1). If one, say 6 X , w i t h o u t loss of general i ty, of the Heegaard curves bi, b2, • • •, bg is p r i m i t i v e i n the h a n d l e b o d y V, t hen a d d i n g 2-handle to V a long the Heegaard curve is a h a n d l e b o d y V of genus g - 1 by G o r d o n ' s T h e o r e m [Gordon] . N o w , (V; b2,---, bg) is a Heegaa rd d i a g r a m o f genus g — 1 of M. 2) . I f c is a s imp le c losed curve i n dV such tha t c is p r i m i t i v e i n V a n d c can be o b t a i n e d by us ing a sequence of connected s u m moves on the curve set b\, b2, • • •, bg, t h e n the connec ted s u m moves give us a Heegaa rd d i a g r a m (V; c, b'2, • • •, b'g). S ince c is p r i m i t i v e , by 1), the Heegaard d i a g r a m (V; c, b'2, • • •, b'g) is reduc ib le . Therefore , the Heegaa rd d i a g r a m ( V ; bi, b2, • • •, bg) is also reducib le . 3) . I f c is a s imp le closed curve i n dV such tha t c is p r i m i t i v e i n V and the reduced w o r d of the curve c co r respond ing to the comple te sys tem {&i, b2, • • • ,bg} of V is 1, t hen by T h e o r e m 2.1, we can use connected s u m moves on the comple te sys t em to o b t a i n c. Therefore , by 2), the Heegaard d i a g r a m ( V ; b\, b2, • • •, bg) is reducib le . • N o w we consider the converse of T h e o r e m 2.3 a n d o b t a i n the fo l l owing resul t . Theorem 2.4 Suppose that (V; bi, b2, • • •, bg) is a Heegaard diagram of genus g of M. If the Heegaard diagram is reducible, then the following conditions are true. 1) . There is a simple closed curve c in dV such that c is primitive in V and c can be obtained by using a sequence of connected sum moves on the complete system {&1.&2, •••,&,} 0fdV. 2) . There is a simple closed curve c in dV such that c is primitive in V and the reduced word of the curve c corresponding to the complete system {b\, b2, • • •, bg} of V is 1. Proof. T h e c o n d i t i o n 1) is equivalent to the c o n d i t i o n 2) by T h e o r e m 2.1. Therefore , we o n l y need to prove the theo rem i n the case of c o n d i t i o n 1). Suppose the Heegaa rd Chapter 2. Geometry and Algebra of Heegaard splittings 17 d i a g r a m (V; 61,6 2, • • • , bg) is reduc ib le a n d (V, W) is the respect ive Heegaard s p l i t t i n g of the Heegaa rd d i a g r a m . T h e n there are p r o p e r l y embedded disks D G V a n d D' E W respec t ive ly such tha t D n D' = 3D Pi 3D' is one po in t , t ha t is , the curve 3D' is a p r i m -i t ive curve o f V. N o t e the w o r d of 3D' co r respond ing to the comple te m e r i d i a n sys tem {61,6 2, • • ' , bg} o f the h a n d l e b o d y W is 1 since 3D' bounds a d i sk D' i n W. Therefore , we can use a sequence o f connected s u m moves o n the comple te sys t em bx,b2, • • •, bg o f 3V to o b t a i n the curve 3D'. • It is a consequence of T h e o r e m 2.4 tha t i f we can find an a l g o r i t h m to o b t a i n a l l p r i m i t i v e curves i n the b o u n d a r y surface of a hand lebody , t hen we have an a l g o r i t h m to detect i f a Heegaard d i a g r a m ( V ; «i , a 2 , • • • , ag) o f a 3 -man i fo ld M is reduc ib le . T h e steps are as fol lows. S tep 1. L i s t a l l p r i m i t i v e curves acco rd ing to the i r w o r d lengths as c i , c 2 , • • • , c m , • • •. S tep 2. F o r i = 1,2, • • •, read the w o r d Wi of c$ co r re spond ing to the comple te sys tem . {bi-, b2l • • • ,bg}. I f Wi is 1, then the Heegaard d i a g r a m is reduc ib le a n d we c a n s top. O t h e r w i s e we cont inue to consider the next p r i m i t i v e curve. If for a sufficiently large N, a l l of the words u>i, w2, • • •, wm are not 1, t hen the Hee-gaa rd d i a g r a m is i r r educ ib le . Note. It is an open p r o b l e m to find a b o u n d N. Example. T h e fo l lowing figure shows a Heegaard d i a g r a m (V; bi, b2) of the 3-sphere [17]. T h e co r re spond ing fundamen ta l group presenta t ion of the Heegaard d i a g r a m is T T ^ S 3 ) =< xi,x2 : x\ • x\,x\ • x\ > . It is c lear tha t c is a s imple closed curve w h i c h satisfies c o n d i t i o n 3) i n the above theorem. Therefore the Heegaa rd d i a g r a m is reducib le . N o w we discuss the a lgebra ic proper t ies of connected s u m moves. Chapter 2. Geometry and Algebra of Heegaard splittings 18 F i g u r e 2.6: A Heegaard d i a g r a m of the 3-sphere T h e words of a s imple c losed curve i n the b o u n d a r y surface of a h a n d l e b o d y corre-s p o n d i n g to different comple te m e r i d i a n systems are i n general comple t e ly different. W e need to k n o w how to choose su i tab le comple te m e r i d i a n systems to make such words s i m -pler . W e k n o w tha t one comple te m e r i d i a n sys tem can be changed to another comple te m e r i d i a n sys t em by us ing connec ted s u m moves ( [37] ). N o w , we invest igate t ransfor-m a t i o n proper t ies of w o r d presentat ions under the connected s u m moves o n a comple te m e r i d i a n systems. Suppose V is a h a n d l e b o d y of genus g w i t h a comple te m e r i d i a n sys tem { d i , • • • , dg} a n d a is a s imp le curve i n dV w h i c h connects two po in t s i n different curves of the comple te m e r i d i a n sys tem, a n d a has no other in te rsec t ion po in t s w i t h the curves i n the comple te m e r i d i a n sys tem. W i t h o u t loss of generali ty, we m a y assume tha t the endpoin t s of a are P i , P 2 a n d Pi £ d\, P2 £ d2. Suppose c is a s imple , or ien ted c losed curve i n dV a n d c intersects a at po in t s Qi,Q2, • • • ,Qn, where we l is t the in te rsec t ion po in t s a cco rd ing to the order o f pass ing t h r o u g h t h e m w h e n we read the w o r d of c. Suppose the w o r d of c Chapter 2. Geometry and Algebra of Heegaard splittings 19 co r r e spond ing to the comple te m e r i d i a n sys tem is wc(dud2,---,dg) = df11**!12 dTdT •••dg2g ••• de1*il4*i2.-.4*i>*i4Vi1+1-"4*1'---d61*al *^aa-.-4*a'ta4Yia+1---4*39---d[knid2kn2 • • • 4n' t n4Y'r+ 1 • • • dgk»° • • • d[mlde2m2 • --dg"19 where 1 < kx < k2 < • • • < kn < m\ G { - 1 , 0 , + 1 } , U G { 1 , 2 , •••,g} for i = 1, 2, • • • , m, j = 1, 2, • • • , g and Qi l ies i n the subarc o f c w h i c h corresponds to the s u b w o r d detka'tsdetks+[s+1 ( or dg^d^1'1 i f the former = g ) of w c for s = 1,2, • • • , n. T h e n we a p p l y the p r o o f of the l e m m a 2 i n the sect ion 2.3 a n d o b t a i n the fo l lowing p r o p o s i t i o n . Proposition 2.1 Suppose we use the sum d[ = di$ad2 ofdx and d2 along a (we assume that their orientations are the same as in Lemma 2.1. ) to replace dx to obtain a new complete meridian system {d[,d2, • • • ,dg}. Then the word of c corresponding to the new system is <K,d2,---,d3) = <o<W12 d[e21(d[d2)e22 d'^id'^)^2 • ••dtl^(d[d'-1fdtl+iek^+1 • • • • • d[ek^(d[d2yk^ • ••dt2^{d\d'-lfdt2+1^^ • --dT29 • • • d i e * B l (d[d2)ek«2 • • • dtne^tn (d^ -YX+i'*1,4"-" • • • dgk»s • • • dxml{d\d2)tm2 •••dgms; where ex,e2,---,ene { - 1 , 1 } . Chapter 2. Geometry and Algebra of Heegaard splittings 20 Proof. W e can d i r e c t l y wr i t e the w o r d w'c(d'x, d2, • • •, da) a cco rd ing to the d iscuss ion i n the p r o o f o f the l e m m a 2. N o t e t h a t ti is equa l to - 1 or 1 is dec ided by the o r ien ta t ions of c a n d d\ near Qi for i — 1,2, • • • , n. • Note. T h e r e is a s i m i l a r resul t for P r o p o s i t i o n 2.1 for any comple te sy s t em o n a n or ien ted surface. B u t the result is i m p o r t a n t to a comple te m e r i d i a n sys t em of a h a n d l e b o d y since the above group t r ans fo rma t ion can be used to present the respect ive t r ans fo rma t ion of presentat ions o f fundamen ta l g roup o f a 3 - m a n i f o l d co r r e spond ing to different bases. ( W e k n o w tha t every base corresponds to a comple te m e r i d i a n sys tem of the respect ive Heegaard hand lebody . ) It is clear tha t connec ted s u m moves on a comple te m e r i d i a n sys tem do not change the p r i m i t i v e p rope r ty of a curve since the p r i m i t i v e p rope r ty is independent of comple te m e r i d i a n systems. 2.2 Dehn twists Somet imes we can no t use connected s u m moves to reduce cance l l i ng pa i rs of words of H e e g a a r d curves. T h e n we consider the use of the D e h n twis ts . T h e w o r d presentat ions of Heegaard curves co r respond ing to a comple te m e r i d i a n sys t em have the fo l l owing t rans format ions under D e h n twis ts . Proposition 2.2 Suppose V is a handlebody of genus g with a complete meridian system {d\, • • •, dg} and D is a properly embedded disk in V. Suppose d — dD and the word Wd of d corresponding to the complete meridian system is wd = d\\-d% dH where ii, i2, • • •, ik G { 1 , 2 , ••-,g} and ei, e2, • • •, ek G { 1 , - 1 } . Suppose c is a simple Chapter 2. Geometry and Algebra of Heegaard splittings 21 closed curve in dV. Suppose the word wc of the curve c corresonding to the system is where j\,j2, • • • ,jn G { 1 , 2 , • • • ,g} and 0:1,0:2, ' • • ,&n £ { 1 , —1}- Suppose that c inter-sects the curve dD at m points Pi, P2, • • •, Pm in order and Ps lies in the subarc of c which corresponds to the subword d^° • d^*®^ of wc and Ps lies in the subarc of 3D which corresponds to the subword d*"° • G^®** of WdD for s = 1, 2, • • • , m, where ps € { 1 , 2 , • • •, n},us G { 1 , 2 , • • •, k} for s = 1,2, • • •, m and Pi < p2 < • • • < pm, and for posi-tive integers p, q, r, we define q@vr as the least positive number with q@pr = q+r(mod p). If we use Dehn twist one time on c which corresponds to a small neighborhood of the disk D in H and is along the direction of 3D, then c is changed to a new curve d whose word wc> corresonding to the system {dx, d2, • • •, dg} is wd = d«l -df2 d j " 1 ' 1 • d^ • derlk\ • <%*>\ •••• •d?2lk1-d?2^C-Proof. W e c a n d i r e c t l y check the move resul t for each s m a l l n e i g h b o r h o o d o f Ps i n the curve c for s = 1,2, • • • , m. T h e s-th s m a l l subarc of c has been changed to a s imp le curve cs w h i c h is j u s t l ike a copy of dD a n d w h i c h has w o r d for s = 1,2, • • • , m. A d d i n g a l l the m words wCl,wC2, • • •, wCm to the w o r d wc i n su i t ab le places a c c o r d i n g to the pos i t ions of respective subarcs , we o b t a i n the w o r d uv whose f o r m is as i n the p r o p o s i t i o n . • Chapter 2. Geometry and Algebra of Heegaard splittings 22 I f we o n l y consider w o r d presentat ions of Heegaard curves, connected s u m moves o n comple te m e r i d i a n systems produce the s i m i l a r results w i t h D e h n twis t . N o w , we note tha t D e h n twis t s do not change the p r i m i t i v e p rope r ty o f a curve i n the b o u n d a r y of a hand lebody . Proposition 2.3 Suppose V is a handlebody of genus g and c is a simple closed curve in dV. Suppose D is a proper embedded disk in V and c' is the curve obtained by using Dehn twist along D on the curve c. Then c is a primitive curve in dV if and only if c' is. • 2.3 Stabilization and reduction A l t h o u g h s t a b i l i z a t i o n a n d r educ t ion are not powerful me thods to s i m p l i f y Heegaa rd d i a -g r a m of a 3 -mani fo ld , they change the genus of a Heegaard d i a g r a m . Therefore somet imes we need i t to o b t a i n m i n i m a l Heegaard d iagrams . ( A minimal or irreducible Heegaard diagram o f a 3 -man i fo ld M is a Heegaard d i a g r a m of M w h i c h can not be reduced, tha t is , does not resul t f rom a s t a b i l i z a t i o n of a Heegaard d i a g r a m of smal le r genus.) A c c o r d i n g to B o i l e a u a n d Zieschang [3], there exists a 3 -man i fo ld w i t h two i r r educ ib le Heegaa rd sp l i t t i ngs , one o f genus 2, the other of genus 3. T h u s m i n i m a l does no t m e a n smal les t genus. T h e t r ans fo rmat ions o f w o r d presenta t ions for s t a b i l i z a t i o n a n d r e d u c t i o n moves are as fol lows: Proposition 2.4 Suppose (V; &i, b2, • • •, bg) is a Heegaard diagram of genus g of M. Suppose we use a stabilization move on this Heegaard diagram to obtain a new Heegaard Chapter 2. Geometry and Algebra of Heegaard splittings 23 diagram (V; b[, b'2, • • •, b'g+1). If the associated fundamental group presentation of the Hee-gaard diagram (V;bx,b2, • • •,bg) is TT(M) = < Xi,x2,---,xg : RuR2, • • •,Rg >, then the associated foundamental group presentation of the Heegaard diagram ( V ; b[, b2, • • •, b'g+1) is TT(M) =< xi,x2, • • •, xg,xg+i : R\,R2, • • •, Rg, xg+\ >. And the word of a curve in 3V is same with the word of the respective curve in dV. Proof. B y the def in i t ion of s t a b i l i z a t i o n . W e need to prove tha t s t ab i l i za t ions have the same p rope r ty as connected s u m moves a n d D e h n twis ts , t ha t is , s t ab i l i za t ions do not change the p r i m i t i v e p rope r ty o f a curve i n the b o u n d a r y of a hand lebody . Proposition 2.5 Suppose (V; bi, b2, • • •, bg) is a Heegaard diagram of genus g of a closed orientable 3-manifold M and c is a simple closed curve in dV. Suppose we use a stabiliza-tion move on this Heegaard diagram to obtain a new Heegaard diagram ( V ; b[, b'2, • • •, b'g+1). If the curve c is a primitive curve for the Heegaard diagram (V; b\, b2, • • •, bg), then the corresponding curve c' of c in dV is a primitive curve for the Heegaard diagram Proof. S ince c is a p r i m i t i v e curve for the Heegaard d i a g r a m (V; bi, b2, • • •, bg), t h e n there is a p rope r embedded d i sk D i n the h a n d l b o d y V such tha t DHc = 3D n c is one po in t a n d we can use connec ted s u m moves i n dV o n the curves &i, b2, • • •, bg t o o b t a i n c. Suppose D' is the respect ive proper e m b e d d i n g d i sk i n V associa ted to D. T h e n clear ly , D' n d = 3D' n d is one po in t i n 3V. N o t e i f we t r y to use connec ted s u m moves o n the curves b'1,b2,-"i b'g to o b t a i n the curve d, there are obs t ruc t ions w h e n we t r y to i s o t o p i c a l l y move a curve pass ing t h r o u g h the disk B w h i c h corresponds to the 1-handle tha t we a d d to V to o b t a i n V. W e solve th is p r o b l e m by us ing the connec ted s u m moves i n the fo l l owing figure. • (V;b[,b'2,---Chapter 2. Geometry and Algebra of Heegaard splittings 24 F i g u r e 2.7: H a n d l e a d d i t i o n move a n d connected s u m move T h u s , we c a n use connected s u m moves on the curves b[, b'2, • • •, b'g+l i n the surface dV to o b t a i n the curve c'. T h e n , we have proven tha t d satisfies the two cond i t i ons of p r i m i t i v e curve i n dV. Therefore , c' is a p r i m i t i v e curve for the Heegaa rd d i a g r a m T h e s i m i l a r resul t for r educ t ion moves is also t rue . Theorem 2.5 Suppose (V; bi, b2, • • •, bg) is a Heegaard diagram of genus g(g > 1) of M and c is a simple closed curve in dV. Suppose D is a properly embedded disk in V such that D D (Uf=1&j) = 0. Suppose cutting V along D has two connected components: One of them is a handlebody V of genus g — 1 and the other connected component is a solid torus T so that bg is a standard meridian curve in dT and T n (ufr^i) = 0. We use a reduction move on the Heegaard diagram (V; b\, b2, • • •, bg) along the disk D to obtain a new Heegaard diagram (V; bi, b2, • • •, 69_i) ( for the sake of simplication, we do not change symbols for the Heegaard curves this time). If c is a primitive curve for the {V'-b1^---• Chapter 2. Geometry and Algebra of Heegaard splittings 25 Heegaard diagram (V; bx,b2,---, bg) and c C dV, then c is also a primitive curve for the Heegaard diagram ( V ; b\, b2, • • •, 6 9 - i ) . Proof. Suppose B is a properly embedded disk in V such that c D B = c D dB is one point. If B n D ^ 0, then each connected component of ( 5 n D ) U (SB D T) bounds a disk in T. Then, we can isotopically move B in the handlebody V such that BnD = $. This indicates that there is a properly embedded disk B' in V such that cDB' = cdB' is one point. On the other hand, since the curve c is a primitive curve for the Heegaard diagram (V; by, b2, • • •, bg), the cyclically reduced word of c corresponding to the curves b\,b2,- • • ,bg is trivial. But c D bg — 0. Then, the reduced word of c corresponding to the curves bi, b2, • • •, is also trivial. The above two conditions indicate that c is a primitive curve for the Heegaard diagram (V';b1,b2,---,bg-i)-• 2.4 A property of moves The following theorem implies that we can reverse the orders between a connected sum move and a Dehn twist. Proposition 2.6 Suppose (V; by, b2, • • •, bg) is a Heegaard diagram of genus g of M. Sup-pose that D is a properly embedded disk in the handlbody V and To is a Dehn twist along D inV. ThenTD(bir--Mcbj,---,bg) = (TD{h), • • • , ^ ( ^ ^ ( ^ ( 6 , ) , • • • ,TD(bg)). That is applying first a connected sum move along an arc c and then a Dehn twist along D is equivalent to applying first the Dehn twist and then a connected sum move along an arc TD (C) . Chapter 2. Geometry and Algebra of Heegaard splittings 26 Proof. IfcHdD — 0, t hen the p r o p o s i t i o n is t rue . If cCidD ^ 0, we m a y assume tha t dc n dD = 0. N o t e tha t i n th i s case the d i sk i n dV g iven by a p p l y i n g the D e h n twis t o n a n e i g h b o r h o o d o f c i n dV is the same w i t h the d i sk w h i c h is a n e i g h b o r h o o d of TD(C) i n dV. • 2.5 Circle and chord presentations of closed 3-manifolds In th i s sec t ion , we in t roduce a new m e t h o d to represent 3-manifolds . W e c a l l th i s m e t h o d the circle and chord presentations of closed 3-manifolds . Definition 2.3 Suppose that (V; bi, b2, • • •, b9) is a Heegaard diagram of M. Suppose that {di, d2, • • •, dg} is a complete meridian system of the handlebody V. Further suppose there are connected curves Ci, c2, • • •, c2g^\ such that the following conditions hold. 1) . For each i G { 1 , 2 , • • •, 2p — 1}, c; C bj for some j G { 1 , 2 , •••,<?} and the intersection points between Ci and the curve set {di,d2, • • - ,dg} are the endpoints of c^. 2) . Cutting dV along the curves di, d2, • • •, dg, c\, c2, • • •, c2g-i is a disk. Then we can use the boundary circle of the disk and some disjoint chords which are the connected components of the set Uf=16j — U 2 ^ 1 ^ to represent the 3-manifold M. We call such a presentation a circle and chord presentation of M. Note. W e can o b t a i n a c i rc le a n d chord presenta t ion f rom each Heegaard d i a g r a m after we i s o t o p i c a l l y move Heegaard curves i n dV. Example. W e c a n o b t a i n a c i rc le a n d chord presenta t ion o f the Po inca r e h o m o l o g y 3-sphere i n F i g . 2.8 by c u t t i n g the surface a long d\, d2, Ci, c 2 , c 3 . F r o m the p reced ing example , we k n o w tha t a c i rc le a n d cho rd presenta t ion of a c losed 3 -man i fo ld consists of a c i rc le a n d some chords. T h e c i rc le consists of the d -curves dx, • • •, dg a n d the c-curves c i , • • • , C 2 9 _ i i n D e f i n i t i o n 2.3. T h e chords consis t of Chapter 2. Geometry and Algebra of Heegaard splittings F i g u r e 2.8: A c i rc le a n d cho rd presenta t ion for the Po inca re h o m o l o g y 3-sphere Chapter 2. Geometry and Algebra of Heegaard splittings 28 the connec ted componen t s of U f = 1 6 j - ( U 2 : ^ 1 ^ ) . A n d we use some m a r k i n g numbers to denote the respect ive in tersec t ion po in t s of the Heegaard curves a n d the curves i n the comple te m e r i d i a n sys tem. O n the o ther h a n d , a g r a p h cons i s t ing o f a c i rc le w i t h some m a r k i n g signs, such as numbers , o n i t a n d some chords m a y cor respond to a Heegaard d i a g r a m of genus g of a c losed 3 -man i fo ld . In the fo l lowing we give cond i t ions for such a l abe l l ed g r a p h to represent a 3 -mani fo ld . Definition 2.4 We classify the marking numbers assigned to vertices in a circle and chord presentation by the following three types. Type I. The marking number which appears four times and each appearance is not an endpoint of any chord. Type II. The marking number which appears three times and among three appearances of such a marking number exactly one is an endpoint of a chord. Type III. The marking number which appears two times and each appearance is an endpoint of a chord. Definition 2.5 An h-arc of a circle and chord presentation is the arc of the circle bounded by two adjacent marking numbers each of which is not endpoint of any chord, m-arcs are the connected components of the complement of all h-arcs in the circle. Example. I n F i g 2.8, we have two m a r k i n g numbers of T y p e 1 ( 2 , 1 1 ) , two m a r k i n g numbers o f T y p e II ( 4, 9 ), n ine m a r k i n g numbers of T y p e III ( 1 , 3, 5, 6, 7, 8, 10, 12, 13 ), s ix h-arcs ( endpoin t s m a r k e d by ( 2, 9 ), ( 11, 2 ), ( 4, 11 ), ( 11, 4 ), ( 2, 11 ), ( 9, 2 ) respec t ive ly ) and s ix m-arcs ( pass ing t h r o u g h ( 9, 10, 11 ), ( 2, 1, 6, 5, 4 ), ( 11, 10, 9, 8, 7, 13, 12, 11 ), ( 4, 3, 2 ), ( 11, 12, 13, 7, 8, 9 ), ( 2, 3, 4, 5, 6, 1, 2 ) respec t ive ly ). N o w , we o b t a i n a necessary a n d sufficent c o n d i t i o n for a g r a p h to cor respond a c i rc le a n d cho rd presenta t ion of a 3 -mani fo ld . Chapter 2. Geometry and Algebra of Heegaard splittings 29 Suppose tha t G is a p l ana r g raph cons i s t ing of a c i rc le a n d dis jo in t chords a n d a l l vert ices of G l ie o n the c i rc le . Suppose tha t the vert ices l i s t ed acco rd ing to the an t i -c lockwise d i r e c t i o n of the c i rc le are P i , P2, • • • , P m a n d Pi has been assigned a m a r k i n g n u m b e r f(Pi) for i = l,2,---,m. T h e n G is a c i rc le a n d chord presenta t ion o f a 3-m a n i f o l d i f a n d o n l y i f i t satisfies the fo l l owing cond i t ions . C o n d i t i o n 1. A l l vert ices are classified by four types: T y p e 1. A ver tex of degree 2 whose m a r k i n g number is o f T y p e I ( i.e., there are exac t ly three other vert ices w i t h the same m a r k i n g number ). T y p e 2. A ver tex of degree 2 whose m a r k i n g number is of T y p e II ( i .e., there are exac t ly two other vert ices w i t h the same m a r k i n g number ). T y p e 3. A ver tex of degree 3 whose m a r k i n g number is of T y p e II ( i .e., there are exac t ly two other vert ices w i t h the same m a r k i n g number ). T y p e 4. A ver tex of degree 2 whose m a r k i n g number is of T y p e I I I ( i .e., there is exac t ly one other ver tex w i t h the same m a r k i n g number ). C o n d i t i o n 2. T h e c i rc le is a u n i o n of two T y p e s of arcs. A n h-arc o n l y inc ludes two adjacent vert ices of degree 2. A n m-arc is a connected componen t of the closure of the complemen t o f a l l h-arcs i n the c i rc le . T w o h-arcs have no c o m m o n endpoin ts a n d for each h-arc b o u n d e d by Pi, Pi®mi, there is exac t ly one h-arc b o u n d e d by Pj, Pj(Bml such tha t / ( P , ) = / ( P , e m i ) , f(Pi®mi) = f(Pj)-F r o m n o w on , we suppose tha t the n u m b e r of h-arcs is ct. T w o m-arcs have no c o m m o n endpoin t s a n d each m-arc does not i nc lude two vert ices w h i c h have the same m a r k i n g number . W e define a n equivalence r e l a t i o n be tween m-arcs as fol lows: T w o m-arcs are m-equivalent i f and o n l y i f a ver tex i n an m-a rc a n d a ver tex i n the o ther m-a rc have the same m a r k i n g number , then we o b t a i n a/2 — 1 equivalence classes. A n d i f we l i s t a l l the m a r k i n g numbers o f the vert ices i n a n equivalence class a c c o r d i n g to an t i -c lockwise d i r ec t i on , we o b t a i n ii, i2, • • •, ik, h, *fc-i>' " ,h, where 2k is Chapter 2. Geometry and Algebra of Heegaard splittings 30 the n u m b e r of the vert ices i n the equivalence class, is G { / ( -P i ) , • • • , f(Pm)} for s = 1 , 2 , - - • , k. F r o m now on , we suppose tha t the number of equivalence classes is g. C o n d i t i o n 3. C o n s i d e r the set cons i s t ing of a l l chords a n d h-arcs. Define an equiva-lence r e l a t i on o n th i s set as fol lows: C a l l two elements i n the set h-equivalent i f they have vert ices whose m a r k i n g numbers are equal . T h e n we have g equivalence classes for th i s equivalence r e l a t i on . C o n d i t i o n 4. A l l chords separate the disk D b o u n d e d by the c i rc le as d is jo in t re-gions. C a l l two such regions D',D" adjacent i f there exist adjacent vert ices P ; , P j © m i G dD' a n d Pj,Pj@ml G dD" such tha t f(PA = / ( P , ) , / ( P i 0 m i ) = / U W ) o r f(pi) = f (Pj®mi), f (Pi®mi) = f(Pj)- "Adjacen t " is an equivalence r e l a t ion . C o n d i t i o n 4 requires tha t a l l regions are i n the same equivalence class. Therefore , we have the fo l l owing theorem. Theorem 2.6 The preceding 4 conditions are necessary and sufficient that a circle and chords with marking endpoints can represent a closed, orientable 3-manifold. Proof. A c c o r d i n g to C o n d i t i o n 2, i f we pa i rwise a t t ach the respect ive vert ices i n an m-equiva lence class w h i c h have the same m a r k i n g number a n d the respect ive arc b o u n d e d by these vert ices a n d pa i rwise a t t ach the two h-arcs w h i c h have the same m a r k i n g number , we o b t a i n a connected , or ientable , c losed surface F of genus g. C o n d i t i o n 3 defines g s imp le c losed curves i n F. C o n d i t i o n 4 ensures the surface a n d curves can be used to present a Heegaard d i a g r a m . ' • T h e fo l l owing theo rem is a c r i t e r i o n to check whether a c i rc le a n d chord presenta t ion correspods to a Heegaa rd d i a g r a m a n d a comple te m e r i d i a n sys tem exists or not . Chapter 2. Geometry and Algebra of Heegaard splittings 31 Theorem 2.7 Suppose ( V ; bi, b2, • • •, bg) is a Heegaard diagram of the 3-manifold M and { d i , d 2 ) • • •, dg} is a complete meridian system of the handlebody V. Let Wi,w2, • • • ,wg be the cyclical words of the respective Heegaard curves associated with the complete meridian system. Then the Heegaard diagram and complete meridian system can give a circle and chord presentation of M if and only if the words wx,w2, • • -,wg include the nontrivial subwords x\\ • xst\, xes\ • xst22, • • • , x^gZ\ • x^Zl; where eu5u e 2 , S2, • • •, e 2 g _ i , <52<,-i € { 1 , - 1 } and x% • xsti + {x% • 4l)5 for i ^ k, i, k e {1 , 2, • • •, 2g - 1}, 6 E {1 , - 1 } . Note. I f Wi = Xk for some i, k 6 { 1 , 2 , • • • ,g}, t hen we regard Wi as i n c l u d i n g the subword xk • xk. Proof. C u t t i n g V a long the curves dx, d2, • • •, dg is a 2-sphere w i t h 2g holes. S u b w o r d xf. • xt\ ind ica tes t ha t there exists a subarc o f the Heegaard curves to connect two holes. T h e c o n d i t i o n xf. • 4; / (xeskk • x5tkk)5 ind ica tes tha t there are no two holes w h i c h are connec ted by two such subarcs. Therefore the u n i o n o f the b o u n d a r y curves o f the holes a n d the subarcs is a connected g raph . F r o m a su i tab le c i rc le a n d chord presenta t ion , we can f o r m some Heegaa rd d i ag rams . It is c lear tha t a l l these Heegaard d i ag rams de te rmine a h o m e o m o r p h i c 3 -man i fo ld . I n fact, a l l these Heegaa rd d i ag rams are equivalent a n d we o n l y need move II to change one to another . T h i s is because i f we let the respect ive s imp le closed curves i n the two comple te m e r i d i a n sys tem m a t c h a n d let the respect ive subarcs i n the circles of the presentat ions m a t c h , a l l o ther pa r t w i l l m a t c h . • Definition 2.6 The complexity of a circle and chord presentation of a 3-manifold M corresponding to a Heegaard diagram ( V ; b\, 6 2 , • • •, bg) and complete meridian system { d i , d 2 , • • •, dp} is defined to be the sum of the lengths of words of the curves bi,b2,---,bg corresponding to the complete meridian system. The genus of such a presentation is the Chapter 2. Geometry and Algebra of Heegaard splittings 32 genus ofV. It is c lear tha t for f ixed pos i t ive integer m, there are o n l y finitely m a n y c i rc le a n d cho rd presenta t ions w i t h c o m p l e x i t y < m for a l l possible 3-manifolds a n d these presentat ions a n d the 3-manifo lds c a n be found. Therefore , we have the fo l lowing theorem. Theorem 2.8 There is an algorithm to list all circle and chord presentations and then to obtain all closed 3-manifolds. There is also an algorithm to list all possible primitive curves for a circle and chord presentation. Proof. B y B i r m a n ' s theorem i n sect ion 1, one can l i s t a l l p r i m i t i v e curves. • Example. T h e 6 c i rc le a n d chord presentat ions w i t h c o m p l e x i t y 7 a n d genus 1 i nc lude a l l lens spaces w i t h fundamenta l g roup Z 7 . Theorem 2.9 There is an algorithm to detect whether a Heegaard diagram is reducible or not. Proof. Suppose the Heegaard d i a g r a m is ( V ; b\, b2, • • •, bg). It is easy to find a c o m -plete m e r i d i a n sys tem {d\, d2, • • •, dg} such tha t a c i rc le a n d cho rd presenta t ion can be ob ta ined . N o w , we c a n d r a w p r i m i t i v e curves a n d check whether we c a n use connected s u m moves o n the Heegaa rd curves bi, b2, • • •, bg to o b t a i n a p r i m i t i v e curve. I f such a p r i m i t i v e curve exists , t hen the Heegaard d i a g r a m is reducib le . • Example. T h e fo l lowing example is a Heegaard d i a g r a m of the 3-sphere ( see F i g . 2.10 ). W e f ind i t is reduc ib le by d r a w i n g respect ive c i rc le a n d cho rd presenta t ion . Chapter 2. Geometry and Algebra of Heegaard splittings 33 F i g u r e 2.9: C i r c l e a n d cho rd presentat ions for the lens spaces L(7, q), q = 1 , 2 , 3 , 4 , 5 , 6 Chapter 2. Geometry and Algebra of Heegaard splittings 34 We know that a Heegaard splitting (V, W) of genus g of M produces a Heegaard dia-gram (V; 61,b2, • • •, bg) where bx, b2, • • •, bg are the respective boundary curves of the disks in a compeltely meridian disk system {By, B2, • • •, Bg} of the handlebody W. Suppose that {dy, d2, • • •, dg} is a complete meridian system of the handlebody V which corre-sponds to a complete meridian disk system {Dx, D2, • • •, Dg} of V. We need to know how to draw the curves di, d 2 , • • • ,dg on the surface dW. The following theorem solves this problem. Theorem 2.10 There exists an algorithm to obtain a circle and chord presentation cor-responding to the Heegaard diagram (W; di, d 2 , • • •, dg), where di = dDi for i = 1,2, • • •, g and the complete meridian system {bx,b2, • • • ,bg} of W directly from a circle and chord presentation corresponding to the Heegaard diagram (V; by, b2, • • •, bg) and the complete meridian system {di, d2,---, dg} of V. Proof. The necessary and sufficient condition that there exists a circle and chord presentation corresponding to the Heegaard diagram (V; 61, b2, • • •, bg) and the complete meridian system {di, d2, • • •, dg} of V is that there does not exist a simple closed curve c C dV such that c n ( u f = 1 ( & j U d j ) ) = 0 and c does not bound a disk in dV. Thus, there exists a circle and chord presentation corresponding to the Heegaard diagram (W; di, d 2 , • • •, dg) and the complete meridian system b2, • • •, bg} of W if and only if there exists a circle and chord presentation corresponding to the Heegaard diagram (V; bi, b2, • • •, bg) and the complete meridian system { d i , d 2 , • • •, dg} of V. Now, suppose the latter exists. To obtain the former, we determine subarcs of the curves d i , d 2 , • • •,dg and use them and the curves h,b2, - • • ,bg to obtain the circle in a circle and chord presentation. This procedure means that we find subarcs c i , c 2, • • •, c2g-i of d i , d 2 , • • •, dg which corresponds to the above subarcs of the curves d i , d 2 , • • •, dg such that cutting V along the simple closed curves h,b2,---,bg and the subarcs Ci, c 2, • • •, c 2 s _ i Chapter 2. Geometry and Algebra of Heegaard splittings Chapter 2. Geometry and Algebra of Heegaard splittings 36 is a d isk . T h e n the o ther par t s of the curves d 1 ? d2, • • •, d9 are the chords i n our c i rc le a n d cho rd presenta t ion . • Definition 2.7 We call the two circle and chord presentations in Theorem 2.10 dual circle and chord presentations. Example. T h e d u a l c i rc le a n d chord presenta t ion of the s t a n d a r d Heegaa rd d i a g r a m of the lens space L(p, q) is jus t the c i rc le and chord presenta t ion of the s t a n d a r d Heegaa rd d i a g r a m of the lens space L(p, q'), where q'q = lmod p. Definition 2.8 Two circle and chord presentations are equivalent if the respective Hee-gaard diagrams corresponding to them are strongly equivalent. Theorem 2.11 Suppose (V, W) and (V',W) are two Heegaard splittings of M. Suppose their respective Heegaard diagrams are (V; b\, b2, • • •, bg) and ( V ; b[, b'2, • • •, b'g). Then the two Heegaard diagrams are equivalent but not strongly equivalent if and only if the circle and chord presentation given by the first Heegaard diagram is equivalent to the dual presentation of the circle and chord presentation given by the second Heegaard diagram. Proof. A d u a l c i rc le a n d chord presenta t ion of the Heegaard d i a g r a m ( V ; b[, b'2, • • •, b'g) corresponds to the Heegaard d i a g r a m (W; d'v d'2, • • •, d'g), where the curves d[, d'2, • • •, d'g consist o f a comple te m e r i d i a n sys tem of the h a n d l e b o d y V. Therefore , acco rd ing to B i r m a n ' s de f in i t ion abou t s t rong equivalences, the theorem is t rue. • Example. T h e c i rc le and chord representat ions of L(7,2) a n d 1/(7,3) d r a w i n g i n th i s sec t ion are d u a l . Therefore , L ( 7 , 2 ) = L ( 7 , 3 ) . I n fact, we can ex tend th is fact to give a new p r o o f of the c lass i f ica t ion theorem of the lens spaces since for q = ±q' Chapter 2. Geometry and Algebra of Heegaard splittings 37 ( m o d p), the c i rc le a n d chord representat ins o f L(p, q) and L(p, q') are same or are same after chang ing the d i r ec t i on of a Heegaard curve; for q • q' = ± 1 ( m o d p ) , the c i rc le a n d c h o r d representat ins o f L ( p , q) a n d L(p, q') are d u a l or are d u a l after chang ing the d i r e c t i o n o f a Heegaard curve. E x a m p l e . A n a p p l i c a t i o n of c i rc le and chord presentat ions is to decide whe the r a w o r d o f a free g roup generated by generators di,d 2 , • • •,d9 is the w o r d of a curve i n the b o u n d a r y surface o f a h a n d l e b o d y o f genus g co r r e spond ing to a comple t e m e r i d i a n sys tem d i , d 2 , • • • , d 9 . F o r example , G i l l m a n [8] gave a ba lanced presenta t ion of the t r i v i a l g roup < a,b : aba~lb~2,6o6_1a-2 > . In th is presenta t ion , the words aba~lb~2,bab~xa~2 c a n no t be the words of the Heegaard curves b\, b2 co r respond ing to a comple te m e r i d i a n s y s t e m { a , b} o f the h a n d l e b o d y V for any Heegaa rd d i a g r a m (V;bi,b2) o f genus 2 o f a c losed 3 -mani fo ld . In fact, we c a n not d raw a curve i n dV w i t h the w o r d aba~lb~2 since the sub-curves ab, ba-1, a _ 1 6 _ 1 gives us a un ique c i rc le i n the c i rc le a n d cho rd presen ta t ion a n d there exists at least one in tersec t ion po in t w h e n we t r y to d r a w the sub-curves b~la. T h i s m e t h o d can be extended to l i s t a l l the words or w o r d sets w h i c h cor respond to a Heegaard d i a g r a m of a 3 -mani fo ld . 2.6 D e t e c t i o n of c losed h o m e o m o r p h i c 3-manifolds I n t h i s sec t ion , we use the c i rc le a n d cho rd presenta t ion to detect t w o h o m e o m o r p h i c 3 -manifolds . W e w i l l analyse the s table equivalence r e l a t ion of two Heegaard d i ag rams step by step. S ince we can de te rmine equivalence of Heegaard d i ag rams t h r o u g h s t rong equivalence o f H e e g a a r d d i a g r a m s a n d the i r d u a l Heegaa rd d i a g r a m s by the las t sec t ion , we w i l l o n l y consider s t rong equivalence. T h r o u g h out th i s sec t ion , we use the fo l lowing def ini t ions . Chapter 2. Geometry and Algebra of Heegaard splittings 38 (1) . M is a connected, c losed, or ientable 3 -mani fo ld . (2) . (V , W), (V, W) are two s t ab ly equivalent Heegaard sp l i t t i ngs of M of genus g. (3) . T h e s t rong equivalent m a p o f the two Heegaa rd s p l i t t i n g is a h o m e o m o r p h i c m a p h : M —> M w i t h h(V) = V, h(W) = W. (4) . W, V, W, V have comple te m e r i d i a n d i sk systems By, B2, • • •, Bg; £>i, D2, • • •, Dg; B[,B'2,---, B'g; D[, D'2, • • •, D'g- respect ively. (5) . bi = dBi- di = dDi- b't = dB'i a n d = dD\ for i = 1, 2, • • • , g. (6) . (V; by, b2, • • •, bg) a n d ( V ; b[, b'2,---, b'g) are the respect ive Heegaa rd d i ag rams of the above two Heegaa rd sp l i t t ings , i.e., bi ( or b\ ) is the image of bi i n the surface dV ( or the i m a g e of b^ i n the surface dV; respect ive ly ) for i = 1,2, • • • , g. (7) . W o r d s of the curves bx,b2,- • • ,bg co r respond ing to the comple te m e r i d i a n sys tem {d[, d'2,---,d'g} are Wi,w2, - • • ,wg respect ively. Case 1. h(Bi) = B'{, h(Di) = D\ for i = 1,2, • • •, g. E v e n i n th is s imp le case, the two Heegaard d i ag rams (V; b\, • • •, bg) a n d ( V ; b[, • • •, b'g) m a y be different. F o r example , we m a y use a D e h n twis t o n V a long a d i sk w h i c h does not intersect the comple te m e r i d i a n d i sk sys tem of V. T h i s does not change the equivalence r e l a t i o n bu t w i l l change the presen ta t ion o f b', • • •, b'g. However , i f we use the c i rc le a n d cho rd presenta t ions a n d choose the same sub-curves for the two Heegaa rd curve systems, t hen the respect ive two ci rc le a n d chord presentat ions w i l l be exac t ly the same since the circles i n the two presentat ions are equal a n d the chords have un ique pos i t ions i n the presenta t ions . Therefore , th i s case w i l l be easi ly dec ided by us ing c i rc le a n d cho rd presentat ions . W e o n l y need to find a l l such presentat ions for the two Heegaa rd d i ag rams a n d compare t h e m . Case 2. h(DA = D[ for i = 1,2, • • •, g. In th i s case, the curves h,b2, • • • ,bg are not the respect ive curves i n the curve set b[, b'2, •• • ,b'g. S ince h(Bx), h(B2), • • •, h(Bg) are disks i n the h a n d l e b o d y W, the curves Chapter 2. Geometry and Algebra of Heegaard splittings 39 h,h, - • • ,bg are connected sums of the curves b[,b'2, • • •,b'g. Then, h,b2, - • • ,bg are con-nected sums of the curves b[, b'2, • • •, b'g. For each circle and chord presentation of the Heegaard diagram (V; b[, b2,---, b'g), we draw the chords which correspond to the words w\, w2, • • •, wg. By the last section, we have only finitely many possible positions for these chords. For each possibility, we read the words of the chord set corresponding to the complete meridian system {b[, b'2,---,b'g}, where the chord set represent the words wi, w2, • • •, wg respectively. Then in this case, at least for one possibility, all the words that we just read have trivial reduced form. Thus, there is an algorithm to determine whether two Heegaard diagrams represent the same manifold so that the equivalent relation belongs to this case or not. Case 3. The general case. In this case, even the curves b\, b2, • • •, bg will not match with the curves b[, b2, • • •, b'g respectively. By the same reason as in Case 2, we know that bi, b2, • • •, bg are connected sums of the curves b[, b'2, • • •, b'g and {b\, b2, • • •, bg} are connected sums of the curves {b[,b'2,---,b'g}. Since we can not directly know the connected sum moves on the curves {b[, b2, • • •, b'g} to obtain the curves {bi, b2, • • •, bg}, we need to consider many complete meridian disk sys-tems of the handlebody V. Suppose that we list all such systems ( according to the num-bers of the intersection points with the original complete meridian system {d\, d2,---, dg} ) as {Dn, • • •, Dig}; {D21, • • •, D2g}; • • •, {Dnl, • • •, Dng}; • • •. Then since the two Hee-gaard diagrams are stably equivalent, there exists a number n such that h(Dni) = D[ for i = l , 2 , T h e remaining problem is to apply the method in Case 2 to find such an n. Combining the Reidemeister-Singer Theorem and the above algorithm, we have the following result. Chapter 2. Geometry and Algebra of Heegaard splittings 40 Theorem 2.12 There is an algorithm to detect if two closed, orientable 3-manifolds are homeomorphic. Note. The method in Case 3 can only detect the stable equivalence property of two Heegaard diagrams if they are really stably equivalent. For two Heegaard diagrams which are not stably equivalent, the above algorithm will not stop since the n does not exist. Therefore, we need to find a bound for n, i.e., we need to find a number M(g) for any two Heegaard diagrams of genus g such that we do not need to consider the cases n> M(g) when we use our algorithm to determine the two Heegaard diagrams are stably equivalent or not. Problem. Find a bound M(g) for the above algorithm. Chapter 3 Heegaard diagrams of the 3-sphere S 3 In this chapter, we will prove that for any two Heegaard diagrams of the same genus of S 3 , there is a sequence of connected sum moves and Dehn twists to pass from one to the other. In particular, there is a sequence of connected sum moves to change the Heegaard curves of a Heegaard diagram (V; bi, b2, • • •, bg) of S 3 to become primitive curves of the handlebody V and there is also a sequence of Dehn twists on the Heegaard curves to change their positions such that the new Heegaard curves bound disjoint disks in R 3 - Int V. Definition 3.1 Let V be a handlebody of genus g in R 3 or S 3 . We say that V lies in R 3 standardly ifV lies in R 3 as in Fig. 3.11. Let the handlebody lie in the R 3 standardly. We know that there exists a simple Heegaard diagram of S 3 for each genus. We call this Heegaard diagram a standard Heegaard diagram of S 3 in R 3 . Definition 3.2 The following Heegaard diagram (V; ex, e2, • • •, eg) of genus g of §3 in R 3 is called the standard Heegaard diagram of genus g of S 3 in R 3 ; where V is the corresponding Heegaard handlebody and e\,e2,---,eg are Heegaard curves. We also call the complete meridian system {di, d2,---, dg} in the figure the standard complete meridian system of the handlebody. One important fact for Heegaard diagrams of § 3 is that we can use a sequence of connected sum moves and Dehn twists to change them to the standard Heegaard diagram 41 Chapter 3. Heegaard diagrams of the 3-sphere S 3 42 F i g u r e 3.12: S t a n d a r d Heegaard d i a g r a m of S 3 Chapter 3. Heegaard diagrams of the 3-sphere S 3 43 of the same genus i f the co r respond ing Heegaard handlebodies l ie i n R 3 s t andard ly . Definition 3.3 Let A be an annulus in 3-space. If there exists a 3-cell B in 3-space such that A C dB, then we say N is attached to the 3-cell B in R 3 . W e also discuss an a l g o r i t h m to de termine Heegaard d i ag rams o f S 3 . 3.1 Simplifying Heegaard diagrams of S 3 W e use D e h n twis t s a n d connec ted s u m moves to change Heegaa rd curves o f any Heegaa rd d i a g r a m of § 3 t o new pos i t ions such tha t the new Heegaard curves have the s imples t forms i n R 3 . Theorem 3.1 Suppose (V; by, b2, • • •, bg) is a Heegaard diagram of genus g o /S 3 . Let the handlebody V lie in a 3-space K 3 or §3 standardly. If the complete system {b\, 62, • • •, bg} consists of primitive curves, then we can use a sequence of Dehn twists and connected sum moves on the complete system {bi, b2, • • •, bg} up to ambient isotopy to obtain the standard Heegaard diagram of genus g o /S 3 . I n the p r o o f o f the theo rem, we w i l l use the fo l lowing l emmas . Lemma 3.1 Suppose V is a handlebody of genus g and {Di, D2, • • •, Dg} is a complete meridian disk system of V and { e i , e2, • • •, eg} is a dual complete system of the system {Di, D2, • • •, Dg} (see Fig. 3.13). Suppose c is a simple curve on dV such that its end-points P,Q lie in the curves 0Ds,dDt respectively, where s,t 6 {1,2, • • • ,g}, s ^ t. If c f i (Uf = 1 6 \Dj ) — {P, Q}, then we can use Dehn twists along some disks which do not intersect the disks in the complete meridian disk system and ambient isotopy on dV to change c to a curve d such that c H ( U f = 1 e j ) = 0. Chapter 3. Heegaard diagrams of the 3-sphere S 3 44 F i g u r e 3.13: A h a n d l e b o d y w i t h two s t a n d a r d comple te systems i n R 3 Note. W e use D e h n twis t s to change the p o s i t i o n of c only, i.e., we keep the curves dDu 0D2, • • •, dDg fixed i n dV. Proof. Deno te the b o u n d a r y curves of the disks D\, D2, • • •, Dg as d l 5 d2, • • •, dg re-spect ively . Suppose the w o r d o f the curve c - b e g i n n i n g f rom the endpo in t P - cor-r e spond ing to the curve sys tem {di,d2, • • •, dg,fei, e2, • • •,eg} is wc = d^ef^ef2 • • • ^dvt\ where /x, v e {1 , - 1 } ; cti,a2, • • •, an G { - 1 , 1 } ; a n d ii,i2, • • • ,in € {1 , 2, • • •,g}. W e w i l l prove t h a t we can use a sequence of D e h n twis ts on the curve c to change i t to a curve d whose w o r d co r re spond ing to the same curve sys tem is df^d^. W e cut the surface dV a long the curves i n the comple te m e r i d i a n sys tem {d1 ? d2,---, dg} a n d o b t a i n a 2-sphere w i t h 2g holes d\, d\, d\, d\, • • •, dlg, d?g, where d\, d| are the two re-spect ive copies o f di for i = 1,2, • • •, g. W e denote th is surface as a n d use A\, A\ to denote the two endpoin t s of ek, k = 1, • • •, g. (See F i g . 3.14) F o r any k G { 1 , 2 , • • •, g} a n d k ^ s, k ^ t, i f c f le k ^ 0, there exists a p o i n t H G c n e j such tha t the p o r t i o n e' of ek b o u n d e d by A\, R satisfies e'Dc = R, i.e., c does not intersect Chapter 3. Heegaard diagrams of the 3-sphere S 3 45 F i g u r e 3.14: C u t t i n g d V a long the comple te m e r i d i a n sys t em the in te r io r of e'. Suppose R separates c in to two s imple curves d, c" whose endpo in t sets are { P , R } , { R , Q } respect ively. T h e n the respect ive words of d,d' co r r e spond ing to the the curve sys tem {e?i, d2, • • •, dg, ely e 2 , • • •, eg} are wd = e ^ e ^ e ? 2 • • • ef™, wc» = e t m ' ' ' e ? n d t > where i m = k. T h e n , c'Ue' is a s imple curve whose endpoin ts P , R l ie o n the curves des, d\ respec t ive ly for some e e {1,2}. N o w , consider a s m a l l regular n e i g h b o r h o o d iV o f g r a p h des U d U d U d\ i n the surface Q. L e t d be the b o u n d a r y connec ted componen t of N not pa ra l l e l to ei ther d\ or d\. T h e n d bounds a d isk D i n the h a n d l e b o d y V. W e use a D e h n twis t a long D i n a sui tab le d i r ec t i on to change c to a curve C\ whose w o r d is wCl = c « e < X 2 ' ' ' C i C '"" C<> w h e r e a G ^ ' W e u s e o n e m o r e D e h n twis t a long D s to cancel the new in tersec t ion po in t between Ci,es. T h u s , the new curve c 2 has at least one less in te rsec t ion po in t s w i t h the curves i n the d u a l comple te sys tem. I n fact, the w o r d of c 2 is wC2 = djfef^e? 2 • • • e ^ " 1 e ^ 1 • • - e ^ d u t w i t h r educ ing some adjacent cance l pa i rs . (See F i g . 3.15) Chapter 3. Heegaard diagrams of the 3-sphere S 3 Figure 3.15: Using two Dehn twists to remove intersection points Chapter 3. Heegaard diagrams of the 3-sphere S 3 47 Applying the preceding construction repeatedly, we may assume that the curve c has been changed to a new curve also denoted by c which does not intersect the curves in the dual complete system {ei , e 2, • • •, eg} except possibly the curves es,et. Finally, we will apply a similar construction as before to cancel the remainning intersection points between c and esUet. If cDet 7^ 0 and Q e d\ in f2, suppose that Z is one of the intersection points between c and et such that c does not intersect with the interior of the simple sub-curve a' of et whose endpoints are A] and Z. Suppose Z separates c into two simple curves also denoted by c',c" whose endpoint sets are {P,Z}, {Z,Q} respectively. Consider a small regular neighborhood U of graph d|Uc yUa'UdJ in the surface f2. The boundary connected components of U consist of three simple closed curves: d], des and another curve d!. d' bounds a disk in V. After we suitably use a Dehn twist on c along d! we change it to a new curve which has one less intersection point with et. This surgery lets the new curve have one more intersection point Y with e s . But Y can be removed by using one Dehn twist move along des as before. There are three other cases we need to consider: c fl et ^ 0 and Q E d] in Q; or c Pi es 7^ 0 and P G d\ in Cl; or c n es ^ 0 and P e d* in Q. It is clear that they can be solved by using the same method as the case we just solved. Therefore, we can use Dehn twists to change c to a new curve which does not intersect with the curves ex, e 2, • • •, eg. • Lemma 3.1 shows how to change the position of c in the surface dV. Now, we consider changes to the curves e i , e 2, • • •, eg that remove the intersection points between c and these curves. Lemma 3.2 Suppose the hypothesis of Lemma 3.1. Then we can use Dehn twists along Chapter 3. Heegaard diagrams of the 3-sphere S 3 48 some disks which do not intersect the curves in the complete meridian system on the curves ex, e 2 , • • •,e9 to obtain a new dual complete system {e\,e'2, • • •,e'g} such that c does not intersect the curves of the new dual complete system. Proof. It follows from Lemma 3.1 that there is a sequence of Dehn twists TX,T2, - • • ,Tn along disks By, B2, • • •, Bn respectively and ambient isotopies which applied to c change c to d such that c 'n ( U f = 1 e ; ) = 0. We also know that the disks Bx, B2, • • •, Bn do not inter-sect the curves dx,d2,---, dg. Now we use a sequence of Dehn twists T~l,T~lx, • • •, T f 1 and ambient isotopies on both d and all the curves in the complete system { e i , • • •, eg}, where T~l is the inverse Dehn twist of T{ for i = 1,2, • • •, n. These moves change d back to c and change the curves ex, e2, • • •, eg to curves e'x, e2, • • •, e'g. d n (uf=1ej) = 0 implies c D (uf= 1ej) = 0 . Since Bx, B2, • • •, Bn do not intersect with the curves dx, d 2 , • • •, dg, the intersections between the curves e'x, e'2, • • •, e'g and the curves dx,d2,---,dg are the same as the corresponding intersections between the curves ex, e 2 , • • •, eg and the curves dx, d 2 , • • •, dg. Therefore, {e'x, e2, • • •, e'g} is a dual complete system of the complete merid-ian system {dx, d2, • • •, dg} • Next, we consider the use of connected sum moves to change a complete meridian system and its dual complete system. Lemma 3.3 Suppose that {dx, d 2 , • • •, dg} is a complete meridian system of a handlebody V of genus g and {bx,b2,---,bg} is a dual complete system, i.e., di n bi is one point Pi and didbk = 0 for i,k = 1,2, • • •, g; i ^ k. Suppose c is a simple curve in dV whose endpoints P,Q respectively lie in the curves ds,dt for some s,t £ {1,2, • • •,g}, s < t and the interior of c does not intersect the curves of the two complete systems. Then {dx, d2,..., ds, • • •, dt-i, ds$cdt, dt+x, • • •, dg} is a complete meridian system of V. It has a dual complete system {bx, b2, • • •,bs-X, bs$c'bt, bs+x, • • • ,bt,- • •, bg}, where we isotopically Chapter 3. Heegaard diagrams of the 3-sphere S 3 49 move P,Q together with their neighbor part sub-curves of c in small neighborhoods of ds U bs, dt U bt in the surface V to obtain the simple curve d whose endpoints lie in bs, bt respectively. Proof, {di, d2,..., ds, • • •, dt-\, ds$cdt, dt+i, • • •, dg} is a complete meridian system of V. Note that the simple closed curve ds$cdt intersects the curves bs,bt at P',Q' re-spectively. P',Q' separate ds$cdt into two simple curves. Denote one of them as d. Then the simple closed curve bs$cibt intersects ds,dt at P, Q respectively and it does not intersect with the curve ds$cdt. Since the interior of c does not intersect the curves dx,---,dg,bx,---,bg, bsic,btn((uf=1di)u(u?=16i)) = (MAnd s )u{b s $ c ,b tndt) = {P,Q} and (ds$cdt) n (Uf=16i) = (ds$cdt) n (6, U bt) = {P',Q'}. Therefore, {di, d2,..., ds, • • •, ds$cdt, dt+\, • • •, dg} has a dual complete system {bi, b2, • • •, bs-i, bs$dbt, bs+i, • • •, bu • • •, bg} and {du ..., ds-U ds$cdu ds+i, • • •, dt, • • •, dg} has a dual complete system {bx, - • •, bs, • • •, bt-\, bs^c>bt, bt+\, • • •, bg}. • Now, we are ready to prove Theorem 3.1. Our main idea is to change a complete meridian system of S 3 to the standard complete meridian system and at the same time to obtain new dual complete systems by using connected sum moves. Proof of Theorem 3.1 Since b\, b2, • • •, bg are primitive curves, there exists a complete meridian system {dx, d2, • • •, dg} of the handlebody V such that bs D dt = 0 and bs D bs is one point Ps for s, t = 1,2, • • •, g, s ^ t , i.e., {h,b2, • • •, bg} is a dual complete system of the complete meridian system. It is easy to see that the reduced words of the curves in a complete meridian system corresponding to another complete meridian system are all trivial. Thus, by the chapter 1, we know that we can use a sequence of connected sum moves on the curves in the first complete meridian system to change them to the curves in the second complete Chapter 3. Heegaard diagrams of the 3-sphere S 3 50 meridian system. Therefore, we can use a sequence of connected sum moves T i , T 2 , • • •, Tm to transform the complete meridian system d2, • • •, dg} to the standard complete meridian system of V. Suppose that d\ = dx,d2 = d2,---,dg0 = dg. Then for i = 1, 2, • • •, ra, Ti is a connected sum move on the curves (Pf1, dl2l, • • •, dl~l which transforms the above curves to the respective curves in a new complete meridian system d\,d2, • • • ,dg, i.e., there are S;, ti e {1, 2, • • •, g}, s; ^ ti and a simple curve Q in dV whose endpoints lie in dl~l,d\~l respectively such that the interior of Cj does not intersect the curves in the complete meridian system {d\~l, dl2~l, • • •, d%~1} and d\. = dt~1^Cidt~1, d\ = d^1 for k e {1,2, • • •, g}, k ^ U; and {d™, d™, • • •, d™} is the standard complete meridian system of V. Now, we use connected sum moves and Dehn twists according to the following steps. Step 1. Use Dehn twists stated in Lemma 3.1 on the curves in the complete system { & i , b2, • • •, bg} to transform them to the curves in a new complete system {b\, b2, • • •, bg} such that c i n (Uf = 16j) = 0. Note that {b\, b\, • • •, bg} still is a dual complete system of the complete meridian system {d l 5 d2, • • •, dg} Step 2. Use one connected sum move 7\ introduced in the above on the curves d\, d2, • • •, dQg to transform them to the curves d\, d\, • • •, d}g. Step 3. Use one connected sum move stated in Lemma 3.3 on the curves in the complete system {b\, b\, • • •, blg} to transform them to curves which form a new complete system {b\, b\, • • •, b2g}. Note that this new complete system is a dual complete system of the complete meridian system {d\, d\, • • •, dlg}. Repeat the above steps on the complete meridian system {d\, d\, - • •, dg} and its dual complete system {b\, b\, • • •, b2g}. Finally, after using the above three steps m times, we change the complete meridian system {d\, d2, • • •, dg} to the standard complete meridian system of V and change the complete system [bi, b2, • • •, bg} to a dual complete system {bf71, b2m, • • •, b2m} of the standard complete meridian system {xx,x2, •• •, xg}. Chapter 3. Heegaard diagrams of the 3-sphere S 3 51 T h a t is j 2 m f l Xj is one po in t i f i = j or is an e m p t y set i f i ^ j for i, j = 1,2, • • •, g. If {of71, blm, • • •, b2m} is not the s t anda rd d u a l comple te sys tem of the s t a n d a r d c o m -plete m e r i d i a n sys tem (see the fo l lowing figure), then we can easi ly find D e h n twis t s o n the curves b2xm, b2m, • • •, b2m a long disks w h i c h do not intersect the d isks i n the s t a n d a r d comple te m e r i d i a n d i sk sys tem to change the curves ft2™, 6 2 m , • • •, b2m to the i r s t a n d a r d pos i t ions one by one. (To move one such a curve, for example & 2 m , to i t s respect ive s t a n d a r d p o s i t i o n ex, we can use a s i m i l a r m e t h o d as i n the p r o o f of the l e m m a 6 to change b2xm t o a curve b'[ w h i c h does no t intersect w i t h the curves e 2 , e 3 , • • • , eg. T h e n we use D e h n twis t s su i t ab le m a n y t imes a long xx to change b'{ to a curve £ x such tha t £ x H ( u f = 1 e j ) = 0, i.e., £ does not intersect w i t h the curves ex, e 2 , • • • , eg, x2, • • •, xg a n d £ intersects w i t h xx at one po in t . Pe rhaps £ s t i l l is not ex. I n th is case, we can use D e h n twis t s a l o n g some disks A i , A 2 , • • • , A m to change the curve £ x to curve £ such tha t £ lies i n a n e i g h b o r h o o d of ex U xx i n dV, where the d i sk A j intersects ex at one po in t a n d A j does not intersect w i t h the curves e 2 , • • • , eg, xx,x2, • • •, xg for j = 1,2, • • • , m. N o w , we can use D e h n twis t s a long xx to change £ to the s t anda rd curve ex. A f t e r we move a 2 ™ to ex, we use the s i m i l a r m e t h o d to move a2m t o e 2 . I t is c lear t h a t w h e n we move y2m, we do no t need to care abou t the curve ex since the curves w h i c h we use D e h n twis t s a long does not intersect w i t h ex. W e cont inue to move the r e m a i n i n g curves to the pos i t ions o f the respect ive curves i n the comple te sys tem { e i , e 2 , • • • , eg} a n d at the same t i m e keep the curves w h i c h are a l ready i n the i r s t a n d a r d p o s i t i o n fixed. F i n a l l y , we change a l l the curves 6 2 m , b2m, • • •, b2m to the i r s t anda rd pos i t ions . ) • Theorem 3.2 Suppose that (V; bx, b2, • • •, bg) is a Heegaard diagram of genus g of S 3 . Then we can use a sequence of connected sum moves and Dehn twists on the Heegaard curves bx,b2,- • • ,bg to transform these curves to the positions of the Heegaard curves of er 3. Heegaard diagrams of the 3-sphere S3 F i g u r e 3.16: U s i n g D e h n twis t s to change curves to s t a n d a r d pos i t ions Chapter 3. Heegaard diagrams of the 3-sphere S 3 53 the standard Heegaard diagram of genus g of S 3 . Proof B y a resul t of W a l d h a u s e n [40], every Heegaard d i a g r a m of genus g of S3 is s t rong ly equivalent to the s t anda rd Heegaard d i a g r a m ( V ; ex, e2, • • •, eg) of genus g of S3. N o t e {&i, b2, • • •, bg} consists o f a comple te m e r i d i a n sys tem of the h a n d l e b o d y Cl(S3 — V), a n d the Heegaard curves ei,e2, - • • ,eg of the s t a n d a r d Heegaa rd d i a g r a m also consis t o f a comple te m e r i d i a n sys tem of the h a n d l e b o d y Cl(S3 — V). S ince these two Heegaa rd d i ag rams are equivalent , there exists a h o m e o m o r p h i s m (f): § 3 —> § 3 such tha t <p(V) = V a n d (f>(Cl(S3 - V)) = Cl(S3 - V). Therefore , ^(ei), <£(e 2 ) , • • • , <t>(eg) b o u n d g pa i rwise d is jo in t d isks i n the h a n d l e b o d y S3 — V a n d c u t t i n g S3 — V a long these d isks is a 3-cel l . Therefore , <j>(ei), 4>(e2), • • • , (j)(eg) consist o f a comple te m e r i d i a n sys t em of S3 — V. N o w , let 61,6 2, • • • , bg be the s t anda rd comple te m e r i d i a n sys tem of V. T h e n by the same reason, {(f>(bi), 4>(b2), • • • , (f>(bg)} is a comple te m e r i d i a n sys t em of V. N o t e {ei, e2, • • • , eg} is the s t a n d a r d d u a l comple te sys tem of the comple te m e r i d i a n sys t em {61, &2, • • •, bg} of the h a n d l e b o d y V. T h e n , 0(ei), (f>(e2), • • • , <j)(eg) consist of a d u a l comple te sys t em of the comple te m e r i d i a n sys tem {(j>(bi),())(b2),-• • ,4>(bg)} of the h a n d l e b o d y V. Therefore </>(ei), (f>(e2), • • • , <j)(eg) consist o f p r i m i t i v e curves i n V. S ince {<fi(ei), <fi(e2), • • • , 4>(eg)} a n d {61, b2, • • •, bg} are two comple te m e r i d i a n systems of the h a n d l e b o d y Cl(S3 — V), the reduced words of the curves bi,b2,---,bg co r r e spond ing to the comple te sys t em (</>(ei), 0(e2), • • • , 4>{e-g)} are t r i v i a l . Therefore , we c a n use a sequence of connec ted s u m moves to change the comple te sys tem {61, b2, • • •, bg} to the comple te sys t em {^(ei), (j)(e2), • • •, (j){eg)} ( T h e o r e m 2.2). Since 4>(ei), (j)(e2), • • •, (j)(eg) consis t of p r i m i t i v e curves i n V, we can use connected s u m moves a n d D e h n twis t s to change t h e m to the pos i t ions of the s t a n d a r d Heegaard curves i n the h a n d l e b o d y V by T h e o r e m 3 . 1 . T h i s completes the p r o o f o f the theorem. • Chapter 3. Heegaard diagrams of the 3-sphere S 3 54 The proof of the above theorem implies the following properties of Heegaard diagrams of S 3 . Corollary 3.1 Suppose that (V; b\, b2, • • •, b9) is a Heegaard diagram of genus g of §3. Then there is a sequence of connected sum moves on the curves bi, b2, • • •, bg to obtain a new Heegaard diagram (V; b[, b2, • • •, b'g) o / S 3 such that b[,b2, • • •, b'g are primitive curves ofV. Proof. In the proof of Theorem 3.2, we have proven that we can use connected sum moves on the curves bx, b2, • • •, bg to obtain primitive curves (f)(ei), (f>(e2), • • •, <j>{eg). • Theorem 3.3 Suppose that (V; bi, b2,---, bg) and (V; b[, b'2,---, b'g) are two Heegaard di-agrams of genus g o /S 3 . Then we can use a sequence of connected sum moves and Dehn twists to pass from one to the other. Proof. By Theorem 3.2, we can use a sequence of connected sum moves and Dehn twists T i , T2, • • •, Tm (or T[, T2, • • •, T'n) to change the Heegaard diagram (V; 6i, b2, • • •, bg) (or the the Heegaard diagram (V; b[, b2, • • •, b'g) respectively) to the standard Heegaard diagram (V; eue2, • • •, eg). Therefore, the moves TUT2, • • • , T m , T ; _ 1 , • • • , T ^ _ 1 , T 1 " 1 will change the Heegaard diagram (V;h ,b 2 , - • • ,bg) to the Heegaard diagram (V;b[,---,b'g)\ where we use T _ 1 to denote the inverse move of a move T. Note the inverse move of a connected sum move (or a Dehn twist) still is a connected sum move (or a Dehn twist respectively). Therefore the corollary is true. • Theorem 3.4 Suppose that (V; b\, b2, • • •, bg) is a Heegaard diagram of genus g o / § 3 and {di, d2,---,dg} is a complete meridian system ofV. Then there is a sequence of connected Chapter 3. Heegaard diagrams of the 3-sphere §3 55 sum moves on the curves bi, b2, • • •, bg to obtain a new Heegaard diagram (V; b[, b2,---, b'g) o / § 3 such that the new Heegaard curves have cyclically reduced words d i , d 2 , • • •,dg cor-responding to the complete meridian system {d\, d 2 , • • •, dg} respectively. Proof. B y T h e o r e m 3.2, we can use connected s u m moves a n d D e h n twis t s to change the Heegaa rd curves bi,b2,---,bg to the d u a l Heegaard curves e\,e2, • • •,eg o f the c o m -ple te m e r i d i a n sy s t em { d i , d 2 , • • • , dg}. N o t e the words o f ei, e2, • • • , eg co r r e spond ing t o the comple te m e r i d i a n sys tem { d i , d 2 , • • • , dg} are d i , d 2 , • • • , dg respect ively. B y L e m m a 11 o f Sec t ion 1.7, we c a n use the co r respond ing connected s u m moves first to o b t a i n a Heegaa rd d i a g r a m (V; b[,b'2, • • •, b'g). T h e n we use the co r re spond ing D e h n twis t s to change the Heegaard d i a g r a m (V\ b[, b'2, • • •, b'g) to the s t anda rd Heegaa rd d i a g r a m . T h e c y c l i c a l l y reduced words o f the Heegaard curves b[, b'2, • • •, b'g co r r e spond ing to the c o m -plete m e r i d i a n sys tem { d i , d 2 , • • • , dg} are { d i , d 2 , • • • , dg} respect ive ly since D e h n twis t s do not change such c y c l i c a l l y reduced words . Remark. W e can not generalize the above co ro l l a ry to two equivalent Heegaard d i a g r a m s of a 3 -mani fo ld . In fact, we can not even general ize i t to two s t rong ly equiv-alent Heegaa rd d i ag rams . W e proved T h e o r e m 3.1 by us ing a very spec ia l p rope r ty o f a Heegaa rd d i a g r a m of S 3 : there exist free p r i m i t i v e curves w h i c h consist of a comple te sys t em of the respect ive hand lebody . F o r o ther 3-manifolds , th i s k i n d of free p r i m i t i v e curves do not exis t . Therefore , the above m e t h o d does not work . F o r example , the s t a n d a r d Heegaard d iagrams co r re spond ing to Lens space L(7,2) a n d L ( 7 , 3 ) are s t ab ly equivalent . B u t we c a n not pass f rom one to the o ther by u s ing connec ted s u m moves a n d D e h n twis ts . T h e reason is tha t the Heegaa rd curve set consists o f one curve we can not use connected s u m moves and i f we use D e h n twis t s o n the s t a n d a r d Heegaa rd d i a g r a m of L ( 7 , 2 ) we w i l l o b t a i n L(7,2 + 7k) for some integer k. Chapter 3. Heegaard diagrams of the 3-sphere S 3 56 Figure 3.17: Two strongly equivalent Heegaard diagrams of (S 1 x §2)jjRP3 Figure 3.17 is another example. Both the Heegaard diagrams in the figure are of (S 1 x S2)jJRP3 and they are strongly equivalent. But we can not use connected sum moves and Dehn twists to pass from one to the other, i.e., in the upper Heegaard diagram, we can not exchange the positions of the two Heegaard curves. 3.2 A n example Example. We consider the example given in Section 1.7. The Heegaard diagram (V;ji,J2,J3) of S 3 is drawn in R3 as Fig. 3.18. The corresponding fundamental group representation of § 3 for this Heegaard diagram Chapter 3. Heegaard diagrams of the 3-sphere S 3 57 Figure 3.18: A Heegaard diagram of S 3 is: TT(S3) = < x,y,z: xyz~zy~lx~l{yzf,xyz'ly~lx~zyz{xyf,xyz~ly~1x~2yz{xyf > . We use a connected sum move on the curves j2,j$ along a (see the above figure), i.e., we use j'2 = J2$ajz to replace j2. Then we obtain a new Heegaard diagram (V; ji, j'2,h) for S 3 . It is easy to see that the curve j'2 is a primitive curve for the handlebody. (See Fig. 3.19) Fig. 3.20 draws the Heegaard diagram after several more Dehn twists. 3.3 Using Dehn twists on Heegaard diagrams of S 3 In Proposition 2.6 of Chapter 2, we proved that we can change the move order of a connected sum move and a Dehn twist without changing the move result. Chapter 3. Heegaard diagrams of the 3-sphere S 3 58 Figure 3.19: Using a connected sum move to simplify the Heegaard diagram Theorem 3.5 Suppose that (V; 6 1 ? 62, • • •, bg) is a Heegaard diagram of genus g of §3. Let the handlebody V lie in R 3 standardly. Then there is a sequence of Dehn twists on the Heegaard curves to obtain a new Heegaard diagram (V; b[, b'2, • • •, b'g) such that the curves b[,b'2,---,b'g bound pairwise disjoint 2-cells in the closure of the complement of V inR3 . First, we prove the following lemma. L e m m a 3.4 Suppose that (V; b\, • • •, bg) is a Heegaard diagram of genus g 0 / S 3 . Let the handlebody V lie in 3-space standardly. Suppose that T is a connected sum move which moves the above Heegaard diagram to a new Heegaard diagram (V; b[, • • •, b'g). Let iVi, • • • Ng, N[, • • •, N'g be small regular neighborhoods of the curves 61, • • •, bg, b[, • • •, b'g in the surface dV respectively. Then if the 2g curves dNx, • • •, dNg form a trivial link in R 3 , then the 2g curves dN^, • • •, dN'g also form a trivial link in R 3 , and vice versa. Chapter 3. Heegaard diagrams of the 3-sphere S 3 59 1. After using Dehn twist on j" 2 along d'. 2. After using Dehn twist on the Heegaard curves along D (D is the disk defined in the picture 1) 3. After use Dehn twist move along B in the B (B is defined in the picture 2) Figure 3.20: Using Dehn twists to simplify the Heegaard diagram Chapter 3. Heegaard diagrams of the 3-sphere S 3 60 Proof. S ince the inverse move of a connected s u m is also a connected s u m move, t hen we can use the inverse move T _ 1 o f T to change the Heegaard d i a g r a m ( V ; b[, 6 2 , • • • , b'g) to the Heegaa rd d i a g r a m ( V ; & i , 6 2 , • • • , bg). I f dN[, dN'2, dN'g is a t r i v i a l l i n k i n 3-space, t hen dNx, dN2, • • •, dNg is a t r i v i a l l i n k i n 3-space too . Therefore , b o t h the curve sets f o r m a t r i v i a l l i n k i n 3-space or b o t h of t h e m do not fo rm a t r i v i a l l i n k i n R 3 . • Proof of Theorem 3.5. B y T h e o r e m 3.2, we c a n use connected s u m moves a n d D e h n twis t s to change the Heegaard curves b\,b2,- • • ,bg to the Heegaard curves ex, e 2 , • • • , eg of the s t a n d a r d Heegaard . Since we can exchange the move order o f one connected- s u m move a n d one D e h n twis t by P r o p o s i t i o n 2.6, then we can use the D e h n twis t s first a n d then use the connec ted s u m moves. T h a t is, i f we suppose these D e h n twis t s change the Heegaa rd d i a g r a m ( V ; 6 1 ( b2, • • •, bg) i n order to Heegaard d i a g r a m ( V ; b'{, b2, • • •, b'g), t h e n after we use the connected s u m moves o n th is new Heegaard s p l i t t i n g i n order we o b t a i n the s t a n d a r d Heegaard d i a g r a m ( V ; e\, e 2 , • • • , eg). Deno te the respect ive s m a l l ne ighborhoods of b'{, &2', • • •, b"g i n dV as J V i , i V 2 , • • •, Ng. I f 2g curves dNudN2,---, dNg do not consis t of a t r i v i a l l i n k i n 3-space, t hen after each connec ted s u m move, the 2g b o u n d a r y curves o f the s m a l l ne ighborhoods of new Heegaard d i a g r a m i n dV do not consist of a t r i v i a l l i n k i n 3-space by L e m m a 3.4. T h i s imp l i e s tha t e i , e 2 , • • • , eg do not b o u n d pa i rwise d is jo in t disks i n the closure of the complemen t of V i n 3-space. T h i s c l ea r ly con t rad ic t s the def in i t ion of the s t anda rd Heegaard d i a g r a m of S 3 . Therefore , dNi, dN2, • • •, dNg is a t r i v i a l l i n k i n 3-space. • W e k n o w tha t every D e h n twis t T twis ts a s m a l l n e i g h b o r h o o d of a d i sk D i n the h a n d l e b o d y V. T h a t is , the d i sk D and twis t d i r ec t i on comple t e ly de te rmine the move. N o t e the c o m p l e x i t y of the d isk D c an be de t e rmined by the l eng th of the w o r d o f dD co r r e spond ing to the s t a n d a r d comple te m e r i d i a n sys t em of V. T o move Heegaa rd curves Chapter 3. Heegaard diagrams of the 3-sphere S 3 61 of a Heegaa rd d i a g r a m to a t r i v i a l l i n k i n R 3 we o n l y need to use some D e h n twis t s a long not too c o m p l i c a t e d disks (i.e., the disks for our D e h n twis t s can be chosen such t ha t t he l e n g t h o f the w o r d o f each o f these d i sks co r re spond ing to the s t a n d a r d comple t e m e r i d i a n sys tem of V is less t h a n a constant k, where k is comple t e ly de t e rmined by the Heegaa rd d i a g r a m ) . Fu r the rmore , g iven a Heegaard d i a g r a m , we t h i n k tha t we o n l y need a cons tant n u m b e r m t imes D e h n twis ts to change the Heegaard curves o f the Heegaa rd d i a g r a m to t r i v i a l l i n k i n R 3 . F o r exmple , perhaps m is less t h a n the cross n u m b e r o f a l l the Heegaa rd curves. I f the above s ta tement is correct , then we w o u l d have a g o o d a l g o r i t h m to de te rmine Heegaa rd d i ag rams o f S 3 . G i v e n a Heegaard d i a g r a m i n R 3 , we c o m p u t e the numbers ra, k first . T h e n we check a l l the possible combina t i ons of n D e h n twis t s to see whe the r we get t r i v i a l l i n k such tha t each move is a long a d i sk w h i c h satisfies tha t the l eng th of the w o r d o f t he d i sk co r re spond ing t o the s t a n d a r d comple te m e r i d i a n sy s t em of V is less t h a n k, where n < m. S ince we o n l y have finite possible combina t ions , th i s a l g o r i t h m c a n finally de te rmine whether the Heegaard d i a g r a m is a Heegaard d i a g r a m of § 3 or not . Chapter 4 Immersing orientable 3-manifolds into R 3 It is w e l l k n o w n tha t every compac t , connected , or ientable 2 -man i fo ld w i t h non -empty b o u n d a r y immerses in to R 2 . T h e i m m e r s i o n can a c t u a l l y be used to descr ibe the 2-m a n i f o l d ( see [27] ). J . H . C . W h i t e h e a d [41] proved tha t a connected , or ien table po lyhe -d r a l 3 -man i fo ld tha t is not closed immerses piecewise l i nea r ly in to R 3 . In th i s chapter , we w i l l give a s t ronger vers ion o f W h i t e h e a d ' s t heo rem i n the spe-c i a l case o f compac t , connected , or ientable 3-manifolds w i t h non -empty b o u n d a r y by c o n s t r u c t i n g an i m m e r s i o n w i t h s ingular i t i es tha t are at most double a n d t r i p l e po in ts . O u r p r o o f uses a Heegaard d i a g r a m . T h e a p p l i c a t i o n of a lgebraic l i n k i n g theory a n d Z 2 - h o m o l o g y i n [41] is rep laced by a d i rec t a n d more t ransparent geometr ic cons t ruc t i on . T h i s geometr ic m e t h o d of p r o o f c a n be adap ted to give a p r o o f of the general case of the W h i t e h e a d theo rem w i t h the a d d i t i o n a l p rope r ty t ha t the s ingular i t i es o f the i m m e r s i o n are double a n d t r i p l e points , only . Theorem 4 . 1 Suppose M is a compact, connected, orientable 3-manifold and dM ^ 0. Then M immerses into R 3 such that the singularities are only double and triple points. I f M is a compac t , connected , or ientable 3 -man i fo ld w i t h dM ^ 0, we embed M i n to a c losed, connected , or ientable 3 -man i fo ld M by a t t a c h i n g a h a n d l e b o d y or a 3-cel l t o each componen t of dM. No te , there is some choice invo lved i n the a t t achment o f a hand lebody , bu t any such M w i l l do. If ( V , W ) is a Heegaa rd s p l i t t i n g o f M a n d ( V ; bg) a Heegaa rd d i a g r a m , let By, • • •, Bg C W be d is jo in t p rope r 2-cells such 62 Chapter 4. Immersing orientable 3-manifolds into R 3 63 t ha t dBi = bi, i = 1, • • •, g. L e t Bi x [—e, e], i = 1, • • •, g, be d is jo in t regular ne ighborhoods of Bi = Bi x 0 i n W w i t h (B{ x [-e, e]) n dW = dB{ x [-e, e], i = 1, • • •, g. C o n s i d e r the p u n c t u r e d m a n i f o l d M 0 = V U ( U f = 1 J 3 j x [-e, e]) w i t h dM0 the 2-sphere [dW - U?=1dBi x ( - e , e)] U (Uf = 173j x ( - e ) ) U ( u f = 1 B j x e). B y the homogenei ty of c losed 3-manifo lds , we m a y assume tha t M c M0 ( there is a 3-cell B3 C M - M a n d there is an ambien t i so topy o n M t ha t moves B 3 to the 3-cell W - (U9i=lBi x ( - e , e ) ) ). A n i m m e r s i o n o f M 0 i n to E 3 w i l l res t r ic t t o a n i m m e r s i o n o f M i n to R 3 . Therefore i t w i l l suffice to prove the theorem for compac t , connected, or ientable 3 -man i fo ld M w i t h dM a 2-sphere w i t h the presenta t ion M = V U (U9=1Bi x [-e, e]). as above. W e use M to denote the co r r e spond ing closed 3 -man i fo ld ob t a ined by a t t a c h i n g a 3-cel l to dM. L e t i : V — > R 3 be an e m b e d d i n g such tha t L(V) is i n the upper h a l f space R 3 of R 3 i n the s t a n d a r d p o s i t i o n F i g . 4.22. W e assume f rom now o n t ha t V = t{V). L e t 7T : R 3 —> R 2 = R 2 x 0 be the p ro jec t ion ir(x, y, z) = (x, y, 0) . Definition 4.1 Let (V; & i , . . . , bg) be a Heegaard diagram of genus g ofM. Let At,..., Ag be small disjoint regular neighborhoods of by,... ,bg in dV. We call Ay,... ,Ag Heegaard annuli of the respective Heegaard curves by,...,bg. Definition 4.2 Let (V; by,..., bg) be a Heegaard diagram of genus g ofM with Heegaard annuli Ay, • • •, Ag. We call the Heegaard curve bi an even ( odd ) Heegaard curve if the following conditions are satisfied: 1) 7T |: Ai —> R 2 is an immersion. 2) The number of self intersection points ofn(bi) is even ( odd respectively ). If all Heegaard curves are even, then we call the Heegaard diagram an even Heegaard diagram. A Heegaa rd d i a g r a m b e i n g even or not even o n l y depends o n the p o s i t i o n o f the Heegaa rd curves i n dV. Chapter 4. Immersing orientable 3-manifolds into R 3 64 W e w i l l prove tha t an even Heegaard d i a g r a m of M defines an i m m e r s i o n of M i n to R 3 . ( Sec t ion 4.2. ) In [24], K i r b y posed the fo l lowing p r o b l e m . P r o b l e m 3.19 of K i r b y ' s p r o b l e m l is t : W h i c h i m m e r s e d 2-spheres i n R 3 b o u n d i m -mersed 3-cells? W e note tha t i n p a r t i c u l a r each even Heegaard d i a g r a m of the 3-cel l defines a n i m -mer s ion of the 3-cel l . W e w i l l prove tha t we can use D e h n twis t s a n d connected s u m moves o n the Hee-gaa rd curves of a Heegaa rd d i a g r a m of M to o b t a i n a n even Heegaa rd d i a g r a m of M ( T h e o r e m 4.3 ). W e also classify a l l compac t , connected , or ientable 3-manifolds w i t h 2-sphere b o u n d -aries w h i c h can be i m m e r s e d in to R 3 w i t h s ingular i t i es tha t are at mos t doub le po in t s . W e w i l l prove the fo l lowing theorem. Theorem 4.2 Suppose M is a compact, connected, orientable 3-manifold with dM con-sisting of 2-spheres. If there exists an immersion r : M — r R 3 whose singulari-ties are at most double points, then M is either a punctured 3-sphere or a punctured ( S 1 x S ^ I J f S 1 x S 2 ) j t - - - J ( S 1 x § 2 ) . Corollary 4.1 Suppose that (V; by, • • •, bg) is. a Heegaard diagram of genus g of a com-pact, connected, orientable 3-manifold M. If M is not homeomorphic to S 3 or ( S 1 x S 2 ) ^ 1 x S 2)Jt • • * JiCS 1 x § 2 ) , then there does not exist an embedding ofV into R 3 such that the boundary of the Heegaard annuli Ax, - •• ,Ag of the respective Heegaard curves form a trivial link in R 3 . A t the end o f the chapter , we give an e x p l i c i t i m m e r s i o n of the p u n c t u r e d rea l p ro -j ec t ive space R P 0 3 i n to R 3 . Chapter 4. Immersing orientable 3-manifolds into R 3 65 4.1 Even Heegaard diagrams We have already defined an even Heegaard diagram. Not all Heegaard diagrams are even. For example, the Heegaard diagram of S 3 = L ( l , 1) is not even. Note that the boundary components of its Heegaard annulus are trivial but linked. ( See Fig. 4.21. ) In this section, we will prove that by applying Dehn twists and connected sum moves to a Heegaard diagram we can obtain an even Heegaard diagram. At first, we isotopically move all Heegaard curves in the boundary of the handlebody into a neighborhood of 3g - 1 standard simple closed curves. Lemma 4.1 Suppose that M is a compact, connected, orientable 3-manifold with bound-ary and (V; by, • • •, bg) is a Heegaard diagram of genus g of M. Assume that the simple closed curves ex, • • •,eg,di, • • • , d 2 9 _ i on the surface dV are as in Fig. 4-22. Let N be a small regular neighborhood of the set U f = 1 e ; U U 2 ^ 1 ^ in dV. We can isotopically move the Heegaard curves bx, • • •, bg in the surface dV to obtain new Heegaard curves b[, • • •, b'g such that the new Heegaard curves lie in N. Proof. Note that dV — IntN is a union of g disks. The Heegaard curves bx, • • •, bg can isotopically be pushed out from these disks into N. • From now on, we assume that all Heegaard diagrams in this section satisfy the con-clusion of Lemma 4.1. Let (V; bx, • • •, bg) be such a Heegaard diagram of genus g of M. Note that n \N is an immersion with at most double points as singularities. It follows Fig. 4.23 that whether the Heegaard diagram (V; bx, • • •, bg) is even or is not even is completely determined by the image of the Heegaard curves bx, • • •, bg under the Chapter 4. Immersing orientable 3-manifolds into R 3 F i g u r e 4.21: A Heegaa rd d i a g r a m of S 3 t ha t is not even Chapter 4. Immersing orientable 3-manifolds into E 3 67 F i g u r e 4.22: Cu rves i n dV t ha t separate dV as d isks p ro j ec t i on 7r. If for k G { 1 , 2 , • • •, g}, s imple closed curve ir(bk) has an even n u m b e r of self-in te rsec t ion po in t s , t h e n the Heegaa rd d i a g r a m is even. Suppose t ha t for k = 1,2, • • • , g, the w o r d wk o f the curve co r respond ing to the curve set {ex, e2, • • •, eg\dx, d2, - • •, dg} is the fo l lowing . W k ~ e a l k a 0 l k ea2k ap2k e<*mkk ap^, where nik, vik G { - 1 , 0 , 1 } for k = 1, • • •, g; i = 1,2, • • •, mk a n d ajk, fyk 6 { 1 , 2 , • • •, </} for k = 1,2, • • • , g\ j = 1,2, • • •, mk. W e assume tha t Hik + v%k > 0. Proposition 4 . 1 Fors = 1,2, •• -,g, letns>k = vik+v2k-\ \-vmkk- Ifnitk,n2,k, • • •,nn>k are all even numbers, then the Heegaard curve ak is even. Proof. T h e fact tha t nSjk is an even number indica tes tha t the curve ck = 7r(a^) has an even n u m b e r of in tersec t ion po in t s w i t h the s imp le closed curve n(ds) i n the xy-plane . N o t e tha t i f ck has an even n u m b e r of in tersec t ion po in t s w i t h 7r(ds) a n d 7r (d s + i ) , Chapter 4. Immersing orientable 3-manifolds into R 3 F i g u r e 4.23: G e t t i n g a d i a g r a m for Heegaa rd a n n u l i Chapter 4. Immersing orientable 3-manifolds into R 3 69 t h e n i t has also an even number of self- intersect ion po in t s w i t h n(ds+g) since c u t t i n g V a l o n g the d isks b o u n d e d by ds, ds+i, ds+g is a 3-cel l for s - 1,2, • • • , g - 1. Therefore , the hypothes i s of the l e m m a imp l i e s tha t ck has an even number of self- intersect ion po in t s i n the xy-plane. T h u s , the Heegaard curve ak is even. • P r o p o s i t i o n 4.1 shows tha t some Heegaard d i ag rams are even. F o r example , the s t a n d a r d Heegaard d i a g r a m of genus 1 of RP3 is even since the Heegaard curve intersects the m e r i d i a n curve i n exac t ly two po in t s ( see F i g . 4.29 ). B u t for the lens space L ( 3 , 1 ) , the s t a n d a r d Heegaard d i a g r a m is not even. However , we can a p p l y a D e h n twis t a long the m e r i d i a n c i rc le to change th is Heegaard d i a g r a m to an even Heegaard d i a g r a m . W e note tha t i n th is case, every t i m e we use a D e h n twis t we a l t e rna t ive ly exchange the o d d / e v e n n u m b e r of the in tersec t ion po in ts . T h u s i n the case of genus 1, we can a lways a p p l y D e h n twis t s to o b t a i n even Heegaard d iagrams . F o r h igher genus cases, u s ing D e h n twis ts alone w i l l i n general not suffice to o b t a i n a n even Heegaa rd d i a g r a m since a D e h n twis t m a y change a n o d d Heegaa rd curve to become even bu t at the same t i m e i t m a y change another even Heegaard curve to become o d d . L e m m a 4 . 2 Fix i e { 1 , 2 , • • • , 2g — 1}. Suppose that di — dDi is the meridian curve in Fig. 4-22 and di intersects a Heegaard curve a at r points. Suppose that along Di we apply a Dehn twist on a to obtain a new Heegaard curve a'. If r is even, then both of a and a' have even numbers of crossings in the box Ri. If r is odd, then a has an even number of crossings in Ri if and only if a' has an odd number of crossings in R and hence a has an odd number of crossings in R if and only if a' has an even number of crossings in R. Proof. In the xy-pl&ne, suppose t ha t m sub-arcs of the curve 7r(a) pass t h r o u g h the b o x R d i rec t ly , n sub-arcs of 7r(a) move a long di once a n d then pass t h r o u g h R a n d k Chapter 4. Immersing orientable 3-manifolds into R 3 70 sub-arcs o f 7r(a) move a r o u n d di once and then do not pass t h r o u g h the b o x ( see the figure a) i n F i g . 4.24 ). T h e number of the crossings o f TT(O) i n Ri is (k+n)(n+m). T h e n , after a D e h n twis t a long di i n the d i r ec t ion of F i g . 4.24, a changes to a ' such tha t 7r(a') has m sub-arcs move a r o u n d di once ( i n the other d i r ec t i on ) a n d then pass t h r o u g h the b o x , n sub-arcs pass t h r o u g h the b o x d i r e c t l y and k sub-arcs of 7r(a) move a r o u n d di once a n d then do not pass t h r o u g h the b o x ( see the figure b) i n F i g . 4.24 ). Therefore , the n u m b e r of the crossings of 7r(a') i n R is (k + m)(n + m). N o t e tha t r = m + n. Therefore , i f r is even, t hen b o t h a a n d a' have even numbers of crossings i n the b o x R. If r is o d d , t hen m is o d d i f a n d o n l y i f n is even. T h u s k + m is o d d i f a n d o n l y i f k + n is even. Therefore , a has an even number o f crossings i n R i f a n d o n l y i f a' has an o d d n u m b e r of crossings i n R. If we use D e h n twis t a long di i n the other d i r ec t i on , t hen the number o f crossings of a' i n R is (k + m + 2n)(n + ra). Since (k + m + 2n)(n + m ) is even i f and o n l y i f (k + m){n + m ) is even, the theo rem is t rue by the preceding proof. • N e x t we consider connected s u m moves. L e m m a 4 .3 Fix i € { 1 , 2 , • • • , 2g — 1} . Suppose that c is a simple curve whose endpoints lie in two Heegaard curves a', a" respectively and Int c f l {a! U a") = 0. Let a = a"$ca" be a connected sum move on a', a" along c. Then a has an even number of crossings in R if and only if both a', a" have even numbers of crossings in R or both of them have odd numbers of crossings in R. Proof. T h e number of crossings o f a i n R is the s u m of the n u m b e r of crossings o f a' i n R, the n u m b e r of crossings of a" i n R a n d the number of crossings i n R p r o d u c e d by the two copies of c used as connected s u m move. T h e two copies of c a lways produces Chapter 4. Immersing orientable 3-manifolds into R 3 71 F i g u r e 4.24: D e h n twis t a long D a n even n u m b e r of crossings. Therefore , the number of crossings o f a i n Ri is even i f a n d o n l y i f the s u m of the number of crossings of a ' i n Ri p lus the number of crossings of a" i n R4 is even. • T h e o r e m 4 .3 Suppose that (V; by, • • •, b9) is a Heegaard diagram of a connected, ori-entable, closed 3-manifold M. Then we can use a sequence of connected sum moves and Dehn twists on the Heegaard diagram to obtain an even Heegaard diagram. Proof. I f each Heegaa rd curve has an even number of crossings i n each box , t hen each Heegaa rd curve is even a n d then the Heegaard d i a g r a m is even. N o w , we assume tha t there exists a box Rj such tha t the Heegaa rd curves bix, • • •, bik have o d d numbers o f crossings i n Ri a n d a l l o ther Heegaa rd curves bik+1, • • • ,big have even numbers o f crossings i n R4. T h e n we replace each bi3 ( s = k + 1, • • • , g ) i n the Chapter 4. Immersing orientable 3-manifolds into R 3 72 Heegaa rd d i a g r a m ( V ; &i,• • •, bg) by a connected s u m b's — bjil^lCsbia to o b t a i n a new Hee-gaa rd d i a g r a m (V; 6 i l 5 • • • , bik, b'k+1, b'k+2, • • •, b'g). N o t e tha t for s = k + 1, k + 2, • • •, g, the new Heegaa rd curve b's has an even number of crossings i n b o x Rj>,f € {1 , - - -, j; — l , j + l , - - - , 2 # — 1} i f and o n l y i f the o l d Heegaard curve bis has an even n u m b e r o f crossings i n the b o x by L e m m a 4.3. N o w , each Heegaard curve i n the new Hee-gaa rd d i a g r a m has an o d d number of crossings i n Rj. W e a p p l y a D e h n twis t a long dj to the Heegaard curves bit, • • •,6ifc, b'k+l, b'k+2, • • • ,b'g t o o b t a i n a Heegaa rd d i a g r a m (V; b'^, • • •, b'ik, bk+1, bk+2, • • •, b"g). T h e n each Heegaard curve of th is Heegaard d i a g r a m has an even n u m b e r o f crossings i n Rj by L e m m a 4.2. S ince our connected moves a n d D e h n twis t s do not change the o d d / e v e n p rope r ty o f the numbers o f crossings i n other boxes, we can use the above m e t h o d to let each Heegaa rd curve have an even number of crossings i n each box . • 4.2 Proof of Theorem 4.1 Definition 4.3 Let K be a simple closed curve in R 3 . An annulus A = K x [—1,1] in R 3 with K = K x 0 is trivial in R 3 if there exists a 2-handle B x [ -1 ,1 ] in R 3 with dB x [ -1 ,1 ] = A. W e need some concepts f rom kno t a n d l i n k theory here. T h e p ro j ec t i on 7r : R 3 —> R 2 = R2 x 0 i n 4.23 defines a regular p ro j ec t i on o n a l i n k L i n R3 cons i s t ing of a l l the Heegaard curves &i, • • • , bg. F o r each cross ing c o f L, 7 r _ 1 ( c ) D L consists o f two po in t s c + = (xc,yc,zc+),c_ = (xc,yc,zc_) w i t h zc+ > zc_. W e say tha t c + is an overcrossing a n d c_ is an undercrossing. T h e segment of L t ha t conta ins the overcross ing or undercross ing of c is ca l l ed the overpass or underpass of c, respect ively. W e w i l l a p p l y the fo l l owing w e l l - k n o w n l e m m a . Chapter 4. Immersing orientable 3-manifolds into R 3 73 Figure 4.25: Changing crossings in a diagram of annuli Lemma 4 .4 Let K be an oriented simple closed curve in B? such that TT(K) in R 2 has only transversal intersection points. We move along K once beginning from a point on K according to its orientation. If for each crossing, we alway pass through the respective overcrossing first, then K bounds a 2-cell in R 3 . • Lemma 4 .5 Let K C R 3 be a knot such that n(K) has only transversal intersection points and their number is even. Let A = K x [—e,e],K = K x 0 be an annulus with 7r |: A —> R 2 = R 2 xO an immersion. Then we can change some crossings of A according to Fig. 4-25 to obtain an annulus A' = K' x [—e, e] such that n(K) = TT(K') and A' is trivial. Proof. If K is a standard trivial knot, i.e., TT(K) is a simple closed curve in the xy-plane, then let K' = K, A' = A. From now on suppose that K is not a standard trivial knot. Chapter 4. Immersing orientable 3-manifolds into R 3 74 F i g u r e 4.26: Re idemeis te r moves L e t P = (zo, 2/0)0) € n(K) satisfy t ha t yo — max{y : (x,y,0) G n{K)} a n d P is not a c ross ing o f K. W e move a l o n g K once b e g i n n i n g f rom 7 r - 1 ( P ) i n R 3 a cco rd ing to the o r i en t a t i on of K. T h a t is, we define a m a p t : [0,1] —> K such tha t t(0) = t(l) = 7 r _ 1 ( P ) a n d t |: (0 ,1) —>• (fC — 7 r _ 1 ( P ) ) is an o r i en ta t ion preserv ing h o m e o m o r p h i s m ( where the o r i en t a t i on o f (0 ,1) is the i n d u c e d o r i en ta t ion of x - ax i s ). S ince K is not a s t a n d a r d t r i v i a l kno t , t hen there exists a\,a]_ € (0 ,1 ) , a\ < o £ such t ha t ir{t(a\)) = 7 r (* (a i ) ) a n d for any 3+,3- G (0,1) w i t h 7r(*(/?+)) = n(t(P-)) we have /?_ > a i . T h a t is c 1 = 7r(t(a^)) = ir(t(a]_)) is the first c ross ing we pass t h r o u g h Chapter 4. Immersing orientable 3-manifolds into R 3 75 both of its overcrossing and undercrossing. Now, let St = {a+, ai, a\, o?_, • • •, a™, a™} be the maximum set such that the fol-lowing conditions are true: 1) . For i = 1,2, • • •, m , a\, ai e (0,1), a\ < ai and 7r(t(a*+)) = 7r(t(ojl_)). That is c1 = Tr(t(al+)) — Tvit^al)) is a crossing of K. 2) . For i, j = 1,2, • • •, m ; i < j, ai < ai and a^., ai £ (a\, ai). That is, after we pass through c\ we do not need to consider the crossings which lie in the closed, passed curve t = n(t(al+,ai)) any more. Then, a\,al_ <E Su i-e., St ^ 0. Now, we change the crossings of K to obtain a knot K' according to the following rules: 1) . If a crossing c ^ {c 1, c 2, • • •, c m } and c + , c_ = £ - 1(7r - 1(c)) with c + < c_, then 7r(c+) should be the overcrossing of c and 7r(c_) should be the undercrossing of c. 2) . For i < m, if i is an odd number, then the overcrossing of Q should be ir(a%+) and the undercrossing of Cj should be 7r(a[_). 3) . For i < ra, if z is an even number, then the overcrossing of c; should be 7T(OJ!_) and the undercrossing of q should be Tf{a\). Note that when we reverse an overpass segment as an underpass segment we always move the underpass segment upward and we assume that the pre-image of the points of the underpass segments for the map t remain fixed ( that is if we move (x, y, z) € K upward to (x, y,z + e), y, z + e)) := t-\(x, y, z)) ). Suppose the above moves change A to an annulus A' in B?. Then A' = K' x [—e, e] and 7r |: A' —> R 2 = R 2 x 0 is an immersion. Further, ir(A') = n(A). Note the projection n gives a diagram K' of the knot K'. Claim 1. We can use a sequence of Reidemeister moves of Type II and Type III on the diagram K' to obtain a knot diagram K" which has only ra crossings c 1, c 2, • • •, cm. Chapter 4. Immersing orientable 3-manifolds into R3 76 Proof of Claim 1. U s i n g Re idemeis te r moves o f T y p e II a n d T y p e I I I o n the respect ive pa r t o f 7r(£ 1) i n K', we can move ir(£l) i n to a s m a l l n e i g h b o r h o o d o f c 1 s ince 7r(^1) is an open arc i n the xy-p\&ne a n d i t o n l y corresponds to overpass segments of K'. N o w , TX{£2) is an open arc i n K' a n d i t o n l y corresponds to overpass segments o f K', thus we can use a sequence o f Re idemeis te r moves of T y p e II a n d T y p e III to cance l a l l crossings i n TT(£2) perhaps except c 1 . W e use th is m e t h o d step by step a n d f ina l ly o b t a i n a kno t d i a g r a m K" w h i c h has o n l y m crossings c 1 , c 2 , • • • , c m . Claim 2. m is an even number. Proof of Claim 2. B y the hypothes is o f the theorem, K has an even n u m b e r o f crossings. K' has the same number of crossings w i t h K. S ince each Re idemei s t e r move of T y p e II does not change the number of crossings a n d each Re idemei s t e r move o f T y p e I I I changes the n u m b e r by 2, by C l a i m 1 imp l i e s tha t m is even. Claim 3. A1 is trivial. Proof of Claim 3. C o n s i d e r the d i a g r a m K" i n C l a i m 1. T h e r e are two cases for the p o r t i o n o f K" i n c l u d i n g c ^ c 2 . ( See a) and b) i n F i g . 4.27. ) B u t Case b) can be moved to Case a) by us ing two Re idemeis te r moves of T y p e II a n d a Re idemei s t e r move o f T y p e I I I . T h e c) a n d d) show tha t the p o r t i o n of A' w i l l be changed to s t a n d a r d case after a t t a c h i n g a 3-cel l to i t . Therefore by i n d u c t i o n , A' is t r i v i a l . • L e m m a 4 .6 Let A = K x [—1,1] be a trivial annulus in R3. Then there are two 2-handles B+ x [ - 1 , 1 ] , B _ x [ -1 ,1 ] with A = (B+ x [ -1 ,1 ] ) n ( £ _ x [ -1 ,1 ] ) = 8(B+ x [ -1 ,1 ] ) n d(£_ x [ -1 ,1 ] ) = 8B+ x [ -1 ,1 ] = 8B+ x [ - 1 , 1 ] . Proof. L e t B+ x [ -1 ,1 ] be a 2-handle w i t h A = 8B+ x [ - 1 , 1 ] . L e t S2 be a 2-sphere i n R 3 a n d let S2 = D+ U D_,D+,D_ 2-cells w i t h 8D+ = 8D_. T h e r e is an ambien t i so topy h t : R 3 —> R 3 , 0 < t < 1 w i t h hi(B+ x 0) = D+. L e t S2 x [ -1 ,1 ] be a regular Chapter 4. Immersing orientable 3-manifolds into R 3 77 F i g u r e 4.27: C h a n g i n g crossings i n a d i a g r a m of a n n u l i n e i g h b o r h o o d o f S2 = S2 x 0. W e m a y assume tha t hx(B+ x [ -1 ,1] ) = D+ x [ - 1 , 1 ] . T h e n S _ x [ -1 ,1 ] = / i _ 1 ( D _ x [ -1 ,1] has the requ i red proper ty . • Theorem 4 .4 Let (V; bx, b2, • • •, bg) be an even Heegaard diagram and A\,A2, • • •, AG its Heegaard annuli. Then there is an immersion r : V —> R 3 with at most double points as singularities such that T(AI),T(A2), • • • ,r(AG) bound disjoint 2-handles in R 3 . Proof. L e t V be embedded i n R 3 as i n F i g . 4.23. T h e n IT | u ? _ i A i is an i m m e r s i o n f r o m the Heegaa rd a n n u l i in to the x y - p l a n e . T h i s i m m e r s i o n ensures t ha t the pos i t ive side of the Heegaa rd a n n u l i is u p w a r d where the pos i t ive side means the outs ide of the h a n d l e b o d y V. T h i s p ro j ec t i on also gives us a d i a g r a m for the a n n u l i A\, A2, • • •, AG i n R 3 . B y L e m m a 4.5, we c a n s u i t a b l y change the crossings i n the annulus d i a g r a m to ob-t a i n a t r i v i a l annulus . T h a t is , we can change the crossings of the Heegaa rd a n n u l i Chapter 4. Immersing orientable 3-manifolds into R 3 78 F i g u r e 4.28: A d d i n g a 3-cell to change cross ing Ai,A2,...,A9 i n R 3 to o b t a i n new a n n u l i A[, A'2,..., A'g such t ha t dA[, dA'2, • • •, dA'g is t r i v i a l i n R 3 . S i m i l a r t o the p r o o f of L e m m a 4.5, we can s u i t a b l y change the crossings p r o d u c e d by different a n n u l i to let Ai l ie above Ai+i for i = 1, • • • , g — 1. N o w , suppose tha t we need to change n c ross ing re la t ionsh ips o f the Heegaa rd a n n u l i w h i c h are near B\, B\\ B\, B\; • • •; B1, B\ respect ively; where B}, Bf is a d i sk p a i r such t ha t two disks n(B}), 7r(B 2) f o r m a cross ing i n the xy-plane a n d B\ lies be low Bf i n the 3-space for i = 1,2, • • • , n. T h a t is, i f we move the par ts of the Heegaa rd a n n u l i near B\, B2, • • •, Bl u p w a r d to let B\ l ie be low B\ i n the 3-space for i — 1,2, • • • , n, t h e n the new a n n u l i are t r i v i a l i n R 3 . T h e above moves can be comple t ed by a t t a c h i n g 3-cells Ei,---,En to B\, • • •, B„ respec t ive ly acco rd ing to F i g 4.28. T h a t is , we a t t ach Ek to B\ a l o n g a d i sk Dk e dEk a n d let the d i sk D'k l ie over the d i sk Bk, where Dk U D'k is a n annulus i n dEk for k = l , 2 , - - - , n . ( S e e F i g . 4.28. ) Chapter 4. Immersing orientable 3-manifolds into E 3 79 N o w , we replace B\, B2, • • •, Bxn by D[, D2, • • •, D'n respect ive ly i n the co r re spond ing a n n u l i to o b t a i n new a n n u l i A[, A'2, • • •, A'g such tha t each annulus is t r i v i a l i n E 3 a n d A\ l ie above A'i+1 for i = 1, • • • , g - 1. . B y L e m m a 4.5, A[, A2, • • •, A'g b o u n d dis jo in t 2-handles. • Proof of Theorem 4-1 B y T h e o r e m 4.4, we can immerse V i n to E 3 w i t h at mos t double po in t s as s ingula r i t i es such tha t the Heegaard a n n u l i can be a t t ached to d is jo in t 2-handles. T h a t is, we can immerse M 0 i n to E 3 w i t h at most double a n d t r i p l e po in t s as s ingu la r i t i e s . • 4.3 Proof of Theorem 4.2 W e w i l l a p p l y the fo l l owing l e m m a . Lemma 4.7 Let S be a set of disjoint 2-spheres and let r : S —> E 3 be an immersion such that T(S) is in general position in E 3 . Ifr has at most double points as singularities, then the image of the singular points consists of disjoint simple closed curves which form a trivial link L in E 3 . The components ofr(S) — L are open disks with holes. Proof. E a c h connec ted componen t of the image of s ingu la r po in t s is a s imp le closed curve since the s ingu la r po in t s are double po in t s a n d the m a p r is an m a p i n general p o s i t i o n . T h e r e exists a s imple closed curve c i n T(S) such tha t c is a connec ted c o m -ponent of the image of s ingu la r po in t s a n d c b o u n d a d i sk D G r(S) so t ha t there are no connec ted componen t s o f the image o f s ingu la r po in t s i n the in te r io r IntD of D, i.e., r |: r _ 1 ( / n t £ ) ) —> IntD is a h o m e o m o r p h i s m . T h e n we m a y use the fo l l owing surgery o n r(S). Suppose tha t D belongs to the image surface Si of r a n d surface S2 is the other image surface w h i c h inc ludes the curve c. C o n s i d e r a s m a l l regular n e i g h b o r h o o d Chapter 4. Immersing orientable 3-manifolds into R3 80 N o f D i n R3 b o u n d e d by the surface 52. W e v i e w J V a s D x [-e, e] w i t h D x 0 = D a n d dD x [—e, e] C S2. W e remove the open annulus dD x (—e, e) f rom S2 a n d a d d the d i sks D x ( - e ) a n d D x e t o S2 - dD x ( - e , e) b y a t t a c h i n g d D x ( - e ) a n d dD x e to the respect ive b o u n d a r y connected components of S2 — dD x (—e, e). T h i s surgery can cance l the s ingu la r connected components of T(S) step by step. N o t e tha t every s imp le c losed curve i n a 2-sphere separates the sphere in to two disks . T h e n , the resul t of the above surgeries is a d is jo in t u n i o n of 2-spheres. Therefore the image of the s ingu la r po in t s consists of d is jo in t s imple closed curves w h i c h f o r m a t r i v i a l l i n k i n R3. I t is c lear t ha t the resul t o f c u t t i n g the image a long the curves is an open d i sk w i t h some closed d i sk removed since each connected componen t of the r e su l t ing surface is a h o m e o m o r p h i c image o f a p o r t i o n of a 2-sphere i n the set S. • N o w , suppose tha t M is a 3 -man i fo ld w i t h dM cons i s t ing o f 2-spheres. Suppose tha t r : M — > R3 is an i m m e r s i o n w i t h at most double po in t s as s ingula r i t i es . T h e n r res t r ic ted to dM is an i m m e r s i o n of dM in to R3. W e m a y assume tha t r(dM) is i n general p o s i t i o n . In the p r o o f o f L e m m a 4.7, there exists a d i sk D so tha t D C r{dM) a n d 3D consists of a connec ted componen t of the image set of the s ingu la r po in t s of the i m m e r s i o n m a p r res t r ic ted to dM a n d there are no other such connected components i n the in te r io r o f the d i sk D. N o t e t ha t T~l{dD) consists of two s imple c losed curves Ci,c2 i n dM such tha t c i bounds a d i sk Dx i n dM w i t h r(Di) = D. L e t the annulus i V be a s m a l l regular n e i g h b o r h o o d of C2 i n dM. W e consider the two cases. C a s e 1. D i s p a r t o f the b o u n d a r y surface o f a connec ted c o m p o n e n t o f the i m a g e set of the s ingu la r po in t s of the i m m e r s i o n m a p r , i.e., the curve C2 bounds a n open d i sk D2 i n the in te r io r of M so t ha t r{D2) — D a n d N is i n the b o u n d a r y of a 3-cel l B w h i c h is Chapter 4. Immersing orientable 3-manifolds into R3 81 a s m a l l regular n e i g h b o r h o o d o f D2 i n M. T h u s B = D2 x [-e, e] w i t h J 5 2 x 0 = D 2 . Case 2. / n i £> does not intersect the image set of the s ingu la r po in t s of the i m m e r s i o n m a p r . In Case 1, we remove D2 x ( - e , e) f rom M a n d o b t a i n the 3 -man i fo ld M' whose b o u n d a r y cont inues to consist of 2-spheres, a n d we res t r ic t the i m m e r s i o n t : M —> R3 to M' —> R3. I n Case 2, we a t t ach a 2-handle B to M a long the annulus TV ( i .e., B = D3 x [-e, e] w i t h dD3 x [-e, e] = N ). T h e resu l t ing 3 -man i fo ld we aga in denote by M'. W e ex tend the i m m e r s i o n t : M — • R3 by h o m e o m o r p h i c a l l y m a p p i n g the 2-hand le B to the co r re spond ing 3-cell B', t ha t is a s m a l l regular n e i g h b o r h o o d of D w i t h r(D) C dB'. B o t h of the above surgeries cance l one connected componen t of the image set o f the s ingu la r po in t s of the i m m e r s i o n r res t r ic ted to dM. T h u s , after finitely m a n y steps we w i l l o b t a i n a 3 -man i fo ld M" w i t h dM" cons i s t ing of 2-spheres a n d an i m m e r s i o n T" : M" —> R3 such tha t the m a p r" res t r ic ted to dM" is a h o m e o m o r p h i s m . Therefore , T" r es t r ic ted to each connected componen t of M" is a h o m e o m o r p h i s m . Suppose M0 is a connec ted componen t of M", t hen r " ( M 0 ) is a compac t , connected 3 -man i fo ld embedded i n E 3 a n d d(r"(Mi)) consists of 2-spheres. Therefore , r " ( M 0 ) is a 3-sphere w i t h holes. T h u s , M" is the d is jo in t u n i o n o f several 3-spheres w i t h holes. N o t e t ha t i n Case 1, we remove a 2-handle f rom M to o b t a i n the 3 -man i fo ld M', a n d i n Case 2, we a d d a 2-handle to M t o o b t a i n the 3 -man i fo ld M'. T h u s we can a d d 1-handles to M" t o o b t a i n a 3 -mani fo ld M* such tha t M C M*. N o t e tha t dM* consists aga in of 2-spheres. Therefore , M* is a p u n c t u r e d 3-sphere or a p u n c t u r e d (S1 x 5 2 ) i l ( 5 1 x S2)i • • • (Sl x S2). S ince M C M* and dM consists o f 2-spheres, t hen M embeds i n Triad x I x I. B y Ro l f sen a n d L i [32], M is a p u n c t u r e d 3-sphere or a p u n c t u r e d ( S 1 x S 2 ) ^ 1 x S^t-'-iS1 x S2). a Chapter 4. Immersing orientable 3-manifolds into R3 82 F i g u r e 4.29: A Heegaard d i a g r a m of RP3 4.4 A n immersion of R P 0 3 , the punctured 3-dimensional projective space, into R3 T h e rea l pro jec t ive p lane R P 2 has a c o m p l i c a t e d i m m e r s i o n in to R3 ( [4], [18], [7], [1] ). N o t e t ha t R P 2 embeds i n R P 0 3 . T h u s the fo l lowing l e m m a gives an i m m e r s i o n of R P in to R 3 . Lemma 4.8 R P 0 3 immerses into R 3 . Proof. R P 0 3 has a Heegaa rd d i a g r a m (V; a) as showing i n F i g . 4.29. A f t e r a d d i n g a 3-cel l D2 x [-e, e] to the so l id torus V = D2 x S 1 by iden t i fy ing 3D2 x [-e, e] w i t h the annu lus N a regular ne ighbo rhood of the Heegaard curve o, we o b t a i n the 3 -man i fo ld R P 0 3 . T h a t is , let h : dD2 x [—e, e] —> N be the i den t i fy ing h o m e o m o r p h i s m , t hen RP3 = VUh (D2x[-e,e\). Chapter 4. Immersing orientable 3-manifolds into R 3 83 N o w , we i s o t o p i c a l l y move V i n R 3 such tha t one side of the annulus N a lways po in t s u p w a r d ( see F i g . 4.30 ). L e t 7r : R3 — • R 2 = E 2 x 0, ir(x,y,z) = (x,y,0), be the pro j ec t i on onto the xy-pl&ne i n E 3 . T h e n TT \N is an i m m e r s i o n . F i g . 4.30 shows tha t the Heegaa rd d i a g r a m is even . A f t e r we change the cross ing acco rd ing to F i g . 4.31, the new annulus is t r i v i a l i n R 3 . Therefore , we can a t t ach a 2-handle to i ts pos i t ive side. • F i g u r e 4.30: A p ro jec t ion f rom R + to the rry-plane Chapter 4. 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