corresponding to those of Heegaard diagrams by the band moves

Transcription

1 Agebra transformatons of the fundamenta groups correspondng to those of Heegaard dagrams by the band moves By Shun HORIGUCHI Abstract. Ths paper gves the basc resut of [1](1997),.e., a hande sdng and a band move of Heegaard dagrams correspond to a repacement and a substtuton n reatons of the fundamenta groups derved from Heegaard dagrams, respectvey (Theorem 12). Coroary 1 s a new addton for the homotopy -sphere. 1. Premnares. Everythng n ths paper, we w be consderng the pecewse near pont of vew. X, Int(X), C(X) shows the boundary, nteror, cosure of a pont set X, respectvey. Hereafter, notaton M denotes a cosed, connected orentabe -manfod uness otherwse stated. In ths secton we gve defntons. We begn wth a defnton of a handebody. Defnton 1. Let { D1,, D n } be mutuay dsonted 2-dsks and h D [0,1] ( 1,, n). A handebody H of genus n s a -ba(cube) B wth n handes { h1,, h n } so that the resut of attachng h wth homeomorphsms throws 2n dsks D 0, D 1 onto 2n dsonted 2-dsks on B n. H s represented as B 1h where B h B h { D 0, D 1}. A handebody H of genus n s aso caed as a sod torus of genus n. We note that H s an orentabe or nonorentabe cosed surface of Euer characterstc 2-2n accordng as H s orentabe or nonorentabe. Defnton 2. Let H be a genus n handebody and { D }( 1,, n), mutuay dsonted propery embedded 2-dsks n H. If the C( H { D1 D n }) becomes a -ba, then the coecton { D }( 1,, n) s caed a compete system of merdan dsks of H and { D }( 1,, n) a compete system of merdan crces of H. Note that { D1,, D n } cuts H nto a 2-sphere wth 2n hoes. Defnton. (1) Let H be an orentabe genus n( 2) n Def. 1. Fg. 1 shows two handes fxed, and sdng D 1 1 handebody wth the same presentaton as h and h of H. By an ambent sotopy of H, keepng D 0 aong the drecton of the ne n turns back to the frst pace. Ths operaton ( B h ), h goes over the h and

2 s caed a hande sdng of h about h. (2) Let { D }( 1,, n) be a compete system of merdan dsks of H and m ( D ) a h h compete system of merdan crces of H. Let α be an arc on H that ons two chosen merdans m, m and Int( ) ( m m ). See Fg. 2. Let N( m m, H ) be a reguar neghborhood of m m on H. D 0 D 1 Fg. 1 B N consst of three crces. Out of the three crces, two are sotopc to m, m and then the remander s not sotopc to them. Let the notaton of remander be m. a band sum of m m s caed and m (wth respect to the h m m h band α). It has aso the very peasant property that bounds a dsk and t s homeomorphc to D and D. Changng the abe m nto m ( m resp.) s caed a band move of m ( m resp.). m Fg. 2 a Defnton 4. A cosed, connected -manfod M s represented wth a unon of two handebodes H 1, H 2 aong ther boundares n M ; M = H 1 H 2 so that H 1 H 2 = H1 H2 H1 H2. H1( H 2) s a cosed surface of genus n( 0). Let the surface be F. H 1 (H 2 resp.) and F are orentabe or nonorentabe accordng as M s orentabe or nonorentabe. A trpet (H 1,H 2,F) or M =H 1 H 2 s caed a Heegaard spttng of M wth genus n and H 1 (H 2 resp.), a Heegaard-handebody. F s caed a Heegaard-surface and the nteger n( 0), Heegaard genus. Let U and V be dsonted handebodes wth the same genus. Let f : U V be a homeomorphsm so that f U : U V s an orentaton-reversng homeomorphsm. Gung together U of U and V of V by f, we obtan M. Then M s denoted as (M ;U,V, f ). It s caed a genus n Heegaard spttng of M concernng f. In (M ;U,V, f ), by repacng f -1 (V) wth V, one can regard (M ;U,V, f ) as (U,V,F) of M. Defnton 5. Suppose ( H1, H2, F) s a genus n( 1) Heegaard spttng of M. Let { D1,, D n },{ D 1,, D n } be a compete system of merdan dsks of H1, H 2, respectvey. Let { m} { m1,, m n } { D1,, D n }, { } { 1,, n} { D 1,, D n }. Then ( H1; m, ) (( H2;, m) resp.) s caed a genus n Heegaard dagram assocated wth ( H1, H2, F). ( m, )((, m) resp.) are caed merdan-ongtude systems of ( H1; m, )(( H2;, m) resp.). Defnton 6. By an ambent sotopy of H, a genus n( 1) handebody H s deformed such as shown n Fg.. 2

3 If a genus n Heegaard dagram ( H ; m, ) satsfes the condtons of 1 m {a pont}( ) and m ( ), then ( H1; m, ) s caed a canonca genus n Heegaard dagram of the -sphere. m 1 m 2 m n 1 2 n (H 1 ; m, ) ( H ; m,, m,,, n ) be a genus n Heegaard dagram assocated wth ( H1, H2, F) of Let 1 1 n 1 Fg. M. We may assume that ( m1 mn ) ( 1 n ) conssts at most of fnte ponts (by an argument of genera poston). Defnton 7. The number of fnte ponts of { m} { } ( m1 mn ) ( 1 n ) s caed a number of cross ponts wth ( H1; m, ) or ( H2;, m). 2. Transformatons of Heegaard dagrams. We begn wth an obvous Proposton. Proposton 8. Let Fg. 4 be a part of Heegaard dagram (U;m,). The ongtude back to crosses the merdan m, turns m and crosses m agan. Then, there exsts a transformaton of (U;m,) so that a part of the dotted ne and t does not cross m. deforms to It does not change the Heegaard genus but decreases the number of cross ponts, as many as 2. h Fg. 4 m Defnton 9. The above transformaton s caed a canceng for a Heegaard dagram. If the dagram ke Fg. 4 appears, then we aways do the above correcton. Let the foowng fgure U-A be a part of Heegaard dagram (U;m,). The ongtudes {,, } ( 0) go around sde by sde on the two handes h and h. The ongtudes 1 {,, },{,, } go around on h, h, respectvey. It shows the genera case that ongtudes 1 p 1 q run on handes h and h. In a speca case that a character on the ower rght equas to 0, there are not ongtudes that run on h and h. V-A s a part of (V;,m), and the dua part of U-A. The ongtude m, m crosses the merdans { 1,, p, 1,, },{ 1,, q, 1,, },

4 respectvey. h m m h 1 p q 1 1 m 1 m 0 1 m B U 1 h 1 h m h p p h h 1 q h q By the hande sdng of U ( B h ), U-B s obtaned from U-A. h about h aong the drectons of the ongtudes { 1,, } n 4

5 1 p h m B U m h q p 1 h 1 h m h p p h 1 1 m h q 1 h 1 q Ths hande sdng s the same as a band sum of 5 m and m wth respect to a part of a ongtude k (1 k ) that exsts between m and m, and a band move of m. In U-B, { 1,, } go around on h (not on h ), { 1,, p } go around on both the h and h, and { 1,, q }

6 do not change the way of runnng. The dua transformaton from V-A nto V-B means that the band move n U-A s carry out n V-A. That s, each merdan k s cut nto two segments by the two ongtudes m and m. Let the shorter segment be α. From m m, we may construct a band sum m of m and m, and carry out a band move of m. Next by an ambent sotopy and reorentng m, V-B s obtaned. It s aso obtaned by hande sdngs of h 1 about { h, h,, h, hp, h,, h }. In V-B, m does not change the way of runnng, 1 and here p1 1 {,,,,, }. The case of (=0), f we can draw a band β m comes to cross 1 q p 1 whch reaches to m 0 va m 1 as t does not ntersect the ongtudes, then we can hande sdng dagram. h about h aong. In ke manners, a hande sdng of h However, ths ony obtans more compex Heegaard h about h, and a band move of m are obtaned. By cancengs for a Heegaard dagram n Def. 9, note that f m, then ongtudes that run between handes h and h s ony a type of Fg. U-A. By the above transformaton we have; Theorem 10. The transformaton from U-A nto U-B s carry out by a hande sdng of h about h, or a band move of band move of m. The dua transformaton from V-A nto V-B s carry out by a m. These transformatons do not change the Heegaard genus but change the number of cross ponts as many as p. In U-A(V-A resp.), we can carry out a band move for two merdans(two ongtudes resp.) m, m n Theorem 10.. Transformatons of the fundamenta groups. To state our resut precsey, we prepare agebra cacuatons for groups. Defnton 11. Let a1,, an r1 1,, rm 1 denotes a presentaton of a fntey generated group, where a1,, an are generators and reator r s a word n the a ' s( 1). We underne to the etters whch are operated. a k Repacements etters; f there are reatons a a w 1( k1,, ) where w k 6 s a word n the ' s( 1), then repace the generator a, etters a a by a new etter a (ths becomes a new generator). Substtuton; f there are two reatons w1 a a where 1 1 and w2 a a 1 a k ( k 1,, ) s a generator and a a s a common word, then substtute 1 a a w1 1 for a a of w2a a

7 Each above agebra cacuaton preserves somorphsm of a group. Let (U,V,F) be a genus n( 1) Heegaard spttng of M and (U;m,) a Heegaard dagram of (U,V,F). { m} { m1,, mn} and { 1,, n} are merdan-ongtude systems. Let each m, orented. By appyng Van Kampen s theorem to U V, of a fundamenta group 1( M ); 1 M m1 m 1ˆ ˆ n n ( ),, 1,, 1 (1) be we may obtan a we-known presentaton We read that m1,, mn are regarded as the generators of the merdans m1,, mn reator ˆ s a word n the turn round m 1 1 m ( resp.) m 1 ' accordng to the orentaton of f s crosses m obtaned by runnng once around the from the eft sde (the rght sde resp.) to the rght sde (the eft sde resp.) of m. See Fg. 5. In the reator ˆ, we may start readng from any m n ˆ because the word ˆ becomes a cycc word by onng both ends of ˆ of etters n ˆ. Therefore ˆ permutatons and nversons. and preservng the sequenta order s unquey defned up to cycc A dua presentaton from (V;, m) of (U,V,F) s aso defned n an anaogous manner, and s denoted as 1 1 n ˆ1, we read the abe ( M ),, m 1,, mˆ 1 (1 ) n and the,.e., whe we take a Group (1) s somorphc to (1 ) but the presentaton s generay dfferent from (1 ) m contnuousy as because merdans and ongtudes are swtched n (U;m,) and (V;,m). Therefore the forms of reators n (1) and (1 ) are dfferent generay. Let a presentaton of the fundamenta group derved from U-A,U-B of (U;m,) be (UA),(UB), respectvey. m +1 m m -1 m, m m m w 1 ( )( k 1,, ) k k 1 k k k m m w 1 ( )( k 1,, p) ( k, ) m w 1 ( )( k 1,, q) r k k =1(reatons other than the above) (UA) m, m m k m w 1 ( )( k 1,, ) k k 1 k k m m w 1 ( )( k 1,, p) ( k, ) m w 1 ( )( k 1,, q) r k k =1(reatons other than the above) (UB) 7

8 Note that the reatons ( )( k 1,, ) n (UA) and (UB), too, do not exst f the ongtudes ( )( k 1,, ) do not exst. k Operatons; n (UA), repace the generator k generator), we get a presentaton that somorphc to (UB). m, etters mm n ( ) by a new etter m (a new Let a presentaton of the fundamenta group derved from V-A,V-B of (V;,m) be (VA),(VB), respectvey.,, 1 p ( ) 1 p 1 m,, 1 q 1 ( ) (VA) 1 q 1 m,, 1 ( k,, ) r =1(reatons other than the above) k k,, 1 p ( ) 1 p 1 m,, 1 q 1 ( ) (VB) 1 q p m 1,, 1 ( k,, ) r =1(reatons other than the above) k Operaton; n (VA), by substtutng = derved from ( m ) 1 p 1 for 1 n ( m ), we get (VB). In ke manner, transformatons of the fundamenta groups correspondng to those of a handes s1dng of h about h of (U;m,) and a band move of m gatherng the Theorem 10 and consderng the above, we have; of (V;,m) are obtaned. Therefore by Theorem 12. The transformaton from U-A nto U-B by a hande sdng of h about h, or a band move of m corresponds to the repacements etters of the fundamenta group. The dua transformaton from V-A nto V-B by a band move of of the fundamenta group. m corresponds to the substtuton Coroary 1. If the fundamenta group derved from a Heegaard dagram of M s transformed nto the trva group by a fnte sequence of the repacement and substtuton correspondng the hande sdng and band move n U-A,V-A, then M References s the -sphere. [1] S. Horguch: Transformatons of the fundamenta groups correspondng to those of Heegaard dagrams by the band moves, Buetn of Ngata Sangyo Unv., No.17, [2] H. Poncaré: Cnquème compément a 1'anayss stus, Rend. Crc. Mat. Paermo 18(1904), 8

9 [] H. Sefert & W. Threfa11: A Textbook of Topoogy, Transated n Engsh by M.A. Godman, Academc Press, nc [4] J.S. Brman: Heegaard sp1ttngs, dagrams and sewngs for cosed orentab1e -manfods, Lecture notes for CBMS conference at Backsburg, (1977). [5] H. Zeschang: Über enfache Kurven auf Vobrezen, Abh. Math. Sem. Unv. Hamburg 25(1962), [6] F. Wadhausen: Heegaard-Zeregungen der -Sphäre, Topoogy 7(1968) [7] J. Hempe1: -manfods, Ann. of Math. Studes 86, Prnceton Unv. Press Shun HORIGUCHI Ngata Sangyo Unversty 470, Karugawa, Kashwazak, Ngata, , Japan E-ma: shor@econ.nsu.ac.p 9