SELF-MAPPING DEGREES OF TORUS BUNDLES AND TORUS SEMI-BUNDLES

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1 Sun, H, Wang, S and Wu, J Osaka J Math 47 (2010), SELF-MAPPING DEGREES OF TORUS BUNDLES AND TORUS SEMI-BUNDLES HONGBIN SUN, SHICHENG WANG and JIANCHUN WU (Receved March 4, 2008, revsed September 24, 2008) Abstract Each closed orented 3-manfold M s naturally assocated wth a set of ntegers D(M), the degrees of all self-maps on M D(M) s determned for each torus bundle and sem-bundle M The structure of torus sem-bundle s studed n detal The paper s a part of a project to determne D(M) for all 3-manfolds n Thurston s pcture Contents 1 Introducton Background Man result Remark on orentaton reversng homeomorphsms Organzaton of the paper Structures of orentable torus bundles and sem-bundles Some elementary facts Classfcatons of torus bundles and sem-bundles Incompressble surfaces Coordnates of torus sem-bundles Lftng automorphsm from sem-bundle to bundle The degrees of self maps of torus bundles The degrees of self maps of torus sem-bundles 149 References Introducton 11 Background Each closed orented n-manfold M s naturally assocated wth a set of ntegers, the degrees of all self-maps on M, denoted as D(M)deg( f ) f Ï M M Indeed the calculaton of D(M) s a classcal topc appeared n many lteratures The result s smple and well-known for dmenson n 1, 2, and for dmenson n 3, there are many nterestng specal results (see [2] and references theren), but t s dffcult to get general results, snce there are no classfcaton results for manfolds of dmenson n Mathematcs Subject Classfcaton 57M10, 55M25

2 132 H SUN, S WANG AND J WU The case of dmenson 3 becomes attractve n the topc and t s possble to calculate D(M) for any closed orented 3-manfold M Snce Thurston s geometrzaton conjecture, whch seems to be confrmed, mples that closed orented 3-manfolds can be classfed n reasonable sense Thurston s geometrzaton conjecture clams that the each Jaco Shalen Johanson decomposton pece of a prme 3-manfold supports one of the eght geometres whch are H 3, PSL(2, R), H 2 E 1, Sol, Nl, E 3, S 3 and S 2 E 1 (for detals see [11] and [10]) Call a closed orentable 3-manfold M s geometrzable f each prme factor of M meets Thurston s geometrzaton conjecture A known rather general fact about D(M) for geometrzable 3-manfolds s the followng: Theorem 11 ([12], Corollary 43) Suppose M s a geometrzable 3-manfold Then M admts a self-map of degree larger than 1 f and only f M s ether (1) covered by a torus bundle over the crcle, or (2) covered by an F S 1 for some compact surface F wth (F) 0, or (3) each prme factor of M s covered by S 3 or S 2 E 1 The proof of the only f part n Theorem 11 s based on the theory of smplcal volume, and varous results on 3-manfold topology and group theory The proof of f part n Theorem 11 s a sequence of elementary constructons, whch were essentally known before Hence for any M not lsted n Theorem 11, D(M) s ether 0, 1, 1 or 0, 1, whch depends on whether M admts a self map of degree 1 or not To determne D(M) for geometrzable 3-manfolds lsted n Theorem 11, let s have a close look of those 3-manfolds from geometrc and topologcal aspects Among Thurston s eght geometres, sx of them belong to the lst n Theorem 11 3-manfolds n (1) are exactly those supportng ether E 3, or Sol or Nl geometres E 3 3-manfolds, Sol 3-manfolds, and some Nl 3-manfolds are torus bundles or sembundles; Nl 3-manfolds whch are not torus bundles or sem-bundles are Sefert spaces havng Eucldean orbfolds wth three sngular ponts 3-manfolds n (2) are exactly those support H 2 E 1 geometry; 3-manfolds supportng S 3 or S 2 E 1 geometres form a proper subset of (3) For 3-manfold M wth S 3 -geometry, D(M) has been presented recently n [1] n term of the orders of 1 (M) and ts elements (and determned earler n [5] when the maps nduce automorphsms on 1 ) Note an algorthm s gven to calculate the degree set of maps between S 3 -manfolds n term of ther Sefert nvarants [8] To determne D(M) for the remanng geometrzable 3-manfolds M, the man task s to solve the queston for the followng three groups (D(M) s rather easy to determne for Sefert manfold M supportng H 2 E 1 or S 2 E 1 geometry): (a) torus bundles and sem-bundles; (b) Nl Sefert manfolds not n (a);

3 SELF-MAPPING DEGREES OF TORUS BUNDLES 133 (c) connected sums of 3-manfolds n (3) do not supportng S 3 or S 2 E 1 geometres Indeed D(M) for M n (a) wll be determned n ths paper (hopefully all the remanng cases wll be solved n a forthcomng paper by the authors and Hao Zheng) 12 Man result In ths paper we calculate D(M) for 3-manfold M whch s ether a torus bundle or sem-bundle To do ths, we need frst to coordnate torus bundles and sem-bundles by nteger matrces n Propostons 13 and 15, then state the results of D(M) n term of those matrces n Theorems 16 and 17 CONVENTION (1) To smplfy notons, for a dffeomorphsm on torus T, we also use to present ts sotopy class and ts nduced 2 by 2 matrx on 1 (T ) for a gven bass (2) Each 3-manfold M s orented, and each 3-submanfold of M and ts boundary have nduced orentatons (3) Suppose S (resp P) s a properly embedded surface (resp an embedded 3-manfold) n a 3-manfold M We use MÒ S (resp MÒP) to denote the resultng manfold obtaned by splttng M along S (resp removng nt P, the nteror of P) A torus bundle s M T I (x, 1) ((x), 0) where s a DEFINITION 12 self-dffeomorphsm of the torus T and I s the nterval [0, 1] For a torus bundle M, we can sotopc to be a lnear dffeomorphsm, whch means ¾ GL 2 () whle not changng M Snce we consder the orentable case only, must be n the specal lnear group SL 2 () Proposton 13 (1) M admts E 3 geometry f and only f s perodcal, or equvalently s conjugate to one of the followng matrces,, , 0 1, 0 1 of fnte order 1, 2, 3, 4 and 6 respectvely; 1 1 (2) M admts Nl geometry f and only f s reducble, or equvalently s conjugate to 1 n where n 0; 0 1 (3) M admts Sol geometry f and only f s Anosov or equvalently s conjugate to a b where a d 2, ad bc 1 c d Proof See [4] DEFINITION 14 Let K be the Klen bottle and N K É I be the twsted I -bundle over K A torus sem-bundle N N N s obtaned by glung two copes along ther torus boundary N va a dffeomorphsm Note N s folated by tor parallel to N wth a Klen bottle at the core of each copy of N

4 134 H SUN, S WANG AND J WU Fg 1 Coordnates of S 1 S 1 I Let (x, y, z) be the coordnate of S 1 S 1 I Then N S 1 S 1 I, where s an orentaton preservng nvoluton such that (x, y, z) (x, y, 1 z), and we have the double coverng pï S 1 S 1 I N Let C x and C y be the two crcles on S 1 S 1 1 defned by y to be constant and x to be constant, see Fg 1 Denote by l 0 p(c x ) (0 slope) and l ½ p(c y ) (½ slope) on N A canoncal coordnate s an orentaton of l 0 l ½, hence there are four choces of canoncal coordnate on N Once canoncal coordnates on each N are chosen, s dentfed wth an element a b of GL c d 2 () gven by (l 0, l ½ ) (l 0, l ½ ) a b c d Proposton 15 Wth sutable choce of canoncal coordnates of N, we have: or 0 1 ;, 0 1 or where z 1 1 z (1) N admts E 3 geometry f and only f 0 1 (2) N admts Nl geometry f and only f z 0; 1 z 0 1 (3) N admts Sol geometry f and only f a b c d Moreover a torus sem-bundle N s also a torus bundle f and only f under sutable choce of canoncal coordnates We wll prove Proposton 15 n Secton 2 Theorem 16 where abcd 0, ad bc 1 z 1 Usng matrx coordnates gven by Proposton 13, D(M ) s lsted n Table 1 for torus bundle M, where Æ(3) Æ(6) 1, Æ(4) 0 Theorem 17 Usng matrx coordnates gven by Proposton 15, D(N ) s lsted n Table 2 for torus sem-bundle N, where Æ(a, d) adgcd(a, d) 2

5 SELF-MAPPING DEGREES OF TORUS BUNDLES 135 Table 1 Degrees of self maps of orentable torus bundles M D(M ) E 3 fnte order k 1, 2 E 3 fnte order k 3, 4, 6 (kt 1)(p 2 Æ(k)pq q 2 ) t, p, q ¾, n 0 l 2 l ¾ Nl n 1 Sol a b c d, a d 2 p 2 (d a)prc br 2 c p, r ¾, ether brc, (d a)rc ¾ or (p(d a) br)c ¾ Table 2 Degrees of self maps of torus sem-bundles N D(N ) E E Nl z 1 Nl 0 1 or 1 z 1 z 0 1 Sol a b, abcd 0, ad bc 1 c d 2l 1 l ¾, z 0 l 2 l ¾, z 0 (2l 1) 2 l ¾ (2l 1) 2 l ¾, f Æ(a, d) s even or (2l 1) 2 l ¾ (2l 1) 2 Æ(a, d) l ¾, f Æ(a, d) s odd 13 Remark on orentaton reversng homeomorphsms Suppose M s a torus bundle or sem-bundle Then any non-zero degree map s homotopc to a coverng ([12] Corollary 04) Hence f 1 ¾ D(M) (whch s computable by Theorems 16 and 17), then M admts an orentaton reversng self homeomorphsm If M s a torus sem-bundle, or M supports the geometry of ether E 3 or Nl, then when M admts an orentaton reversng self homeomorphsm s explctly presented n the followng: Corollary 18 (1) A torus sem-bundle N admts an orentaton reversng homeomorphsm f and only f s ether, or 0 1, or a b where abc c a (2) A torus bundle M supportng E 3 geometry admts an orentaton reversng homeomorphsm f and only f s ether, or 0 1 (3) If M supports Nl geometry, then M admts no orentaton reversng homeomorphsm For torus bundle wth gven Anosov monodromy, even we can calculate whether 1 ¾ D(M ), but there seems no smple descrpton as n Corollary 18 (The referee

6 136 H SUN, S WANG AND J WU nformed us that there s a convenent descrpton of when 1 ¾ D(M ), see Lemma 17, [9]) EXAMPLE 19 For the torus bundle M, a b c d, 1 ¾ D(M ) Indeed for, f a d 3, then 1 ¾ D(M ) Snce p 2 (d a)prc br 2 c 1 has soluton p 1 d, r c when a d 3, and soluton p 1 d, r c when a d 3 EXAMPLE 110 For the torus bundle M, 2 3 a b c d, f ad 2 has prme decomposton p e p e n n, 1 D(M ) Indeed for such that p 4l3 and e 2m 1 for some, then 1 D(M ) Snce f the equaton p 2 (d a)prc br 2 c 1 has nteger soluton, (((a d) 2 4)r 2 4c 2 )c 2 should be a square of ratonal number That s ((a d) 2 4)r 2 4c 2 s 2 for some nteger s Therefore (a d 2)(a d 2)r 2 s a sum of two squares By a fact n elementary number theory, nether a d 2 nor a d 2 has 4k 3 type prme factor wth odd power (see p 279, [7]) EXAMPLE 111 the torus bundle M, 2 1 9, 11, 16, 19, 20 ¾ D(M ) Note f 1 ¾ D(M), then k ¾ D(M) mples k ¾ D(M) For 1 1, among the frst 20 ntegers 0, exactly 1, 4, 5, 14 Organzaton of the paper Theorems 16 and 17 wll be proved n Sectons 3 and 4 respectvely To prove these theorems, we need have a careful look of the structures of torus bundle and sem-bundles Ths s carred out n Secton 2 We explan more about Secton 2 The most convenent and useful reference for us s Notes on basc 3-manfold topology by Hatcher [4], whch s not formally publshed, but wdely crculated (see In partcular Chapter 2 of [4] s devoted to the study of torus bundles and sem-bundles Theorems 23 and 24 about classfcatons of torus bundles and sem-bundles are quoted from [4] drectly It seems that the proof of Theorem 24 n [4] mssed an exsted and rather complcated case, so we rewrte a proof for t (most parts stll follow that n [4]) Lemma 26 studes ncompressble surfaces n torus sem-bundle, whch reles on the proof of Theorem 24 Then Proposton 15 s proved by usng Theorem 24, Lemma 26, and Lemma 28 whch presents the relaton between glung maps of a torus sem-bundles and ts torus bundle double covers Fnally, Theorem 29 studes lftng of maps between torus sem-bundles to ther torus bundle double covers 2 Structures of orentable torus bundles and sem-bundles 21 Some elementary facts All facts n ths sub-secton are known, and one can fnd them n [6], or more drectly n [4]

7 SELF-MAPPING DEGREES OF TORUS BUNDLES 137 Fg 2 Coordnates of N DEFINITION 21 Suppose an orented 3-manfold M ¼ s a crcle bundle wth a gven secton F, where F s a compact surface wth boundary components c 1,, c n,, c nb wth n 0 On each boundary component of M ¼, orent c and the crcle fber l so that the product of ther orentatons match wth the nduced orentaton of M ¼ Now attachng n sold tor S to the frst n boundary tor of M ¼ so that the merdan of S s dentfed wth slope r a c b l wth a 0 Denote the resultng manfold by M whch has the Sefert fber structure extended from the crcle bundle structure of M ¼ We wll denote ths Sefert fberng of M by M( g, bá r 1,, r s ) where g s the genus of the secton F of M, wth the sgn f F s orentable and f F s nonorentable, here genus of nonorentable surfaces means the number of R P 2 connected summands When b 0, call e(m) È s 1 r the Euler number of the Sefert fberaton Another vew of N descrbed n Fg 2 (a): N s obtaned from S 1 I I by dentfyng S 1 I 0 wth S 1 I 1 va a dffeomorphsm whch reflects both the S 1 and I factors Fg 2 (b) s a schematc pcture of N whch wll be used n the paper We lst some propertes of N as: Lemma 22 (1) N has two types of Sefert fber structures: I: M(0, 1Á12, 12) n whch l 0 on N s a regular fber and l ½ s the boundary of the secton defnng the Sefert nvarant II: M( 1, 1Á ) n whch l ½ on N s a regular fber and l 0 s the boundary of the secton defnng the Sefert nvarant (2) N has three types of essental (orentable, ncompressble, -ncompressble) surfaces: I A torus parallel to N II An annulus whose boundary s l ½ n N (Fg 3 (a)) whch does not separate N III An annulus whose boundary s l 0 n N (Fg 3 (b)) whch separates N (3) Suppose M s a torus bundle or sem-bundle and F s a closed ncompressble surface n M, then F s unon of parallel tor

8 138 H SUN, S WANG AND J WU Fg 3 Essental surface n N 22 Classfcatons of torus bundles and sem-bundles Orentable torus bundles and sem-bundles are classfed by two theorems below Theorem 23 ([3]; [4], Theorem 26) An orentable torus bundle M s dffeomorphc to M f and only f conjugates to 1 n GL 2 () Theorem 24 ([4], Theorem 28) The torus sem-bundle N s dffeomorphc to N f and only f of sgns understood n GL 2 (), wth ndependent choces Proof (We start the proof as that n [4]) Suppose f Ï N N s a dffeomorphsm and T, T ¼ are the torus fbers of N, N respectvely N Ò T ¼ N 1 N 2 where N 1, N 2 are homeomorphc to N Snce f s a dffeomorphsm, two components of N Ò f (T ) are both homeo- s an es- morphc to N We can sotope f, such that every component of f (T ) N sental surface n N, 1, 2 So f (T ) N s n the three types lsted n Lemma 22 (2) Thus ether f (T ) s parallel to T ¼, or takes l 0 or l ½ to l 0 or l ½ Suppose f (T ) s parallel to T ¼ We can assume f (T ) T ¼ Then must be obtaned from by composng on the left and rght homeomorphsms of N whch extend to homeomorphsms of N Such homeomorphsms must preserve both l 0 and l ½ (may reverse the drectons), snce l 0 s the unque slopes of the boundares of essental separatng annulus and l ½ s the unque slopes of the boundares of essental non-separatng annulus n N Theorem 24 s proved n ths stuaton Suppose takes l 0 or l ½ to l 0 or l ½ Then there are three cases as below: CASE (1) takes l ½ to l 0 (f takes l 0 to l ½, then we consder 1 ) CASE (2) takes l ½ to l ½ CASE (3) takes l 0 to l 0 (The proof n [4] clams that only Case (3) s possble, whle we show below that only Case (2) s mpossble) Case (1) Now = z 1, and N M( 1, 0Á12, 12, z), and e(m)z Note:

9 SELF-MAPPING DEGREES OF TORUS BUNDLES 139 Fg 4 Cut N through type (II) surfaces Fg 5 Cut N through type (III) surfaces () f (T ) N 1 are n parallel annul A 1,, A n of type (II) (see Fg 4), whch are located n a cyclc order n N Set A a a ¼, then 2n crcles a 1,, a n, a ¼ 1,, a¼ n are located n cyclc order n N 1 () f (T ) N 2 are annul B 1,, B n of type (III) (see Fg 5), where B 1 s next to B, 1,, n 1 n N 2 Set B b b ¼ then 2n crcles b 1,, b n, b ¼ n,, b¼ 1 are located n cyclc order n N 2 If n 1, we can check that pastes A 1 and B 1 to a Klen bottle, whch contradcts the fact that f (T ) s torus When n 1, we can assume pastes a 1 to b 1 and pastes a 2 to b 2, after rendexng A f necessary By the orders of sequences of a 1,, a n, a ¼ 1,, a¼ n and b 1,, b n, b ¼ n,, b¼ 1 on N 1 and N 2, we have a s pasted to b, and a ¼ pasted to b ¼ n, 1,, n So A, A n, B, B n are pasted to one component of f (T ) n N, and f (T ) has [(n 1)2] components Snce f (T ) s connected, we have n 2 Now N 1 Ò f (T ) can be presented as two I -bundles over annulus: I A 1 and I A 2, where f (T ) N 1 A 1 A 2, as n Fg 4 N 2 Ò f (T ) can be presented as an I -bundle

10 140 H SUN, S WANG AND J WU Fg 6 over annulus I B as n Fg 6 (a) and two sold tor P 1 and P 2 wth the core of P N 2 to be the (2, 1) curve of P as n Fg 6 (b) If we glue those fve peces along N, we get two components of N Ò f (T ) whch f needed),, then 2n crcles are N1 ¼ P 1 N I A 1 N P 2 and N2 ¼ I A 2 N I B (re-ndex A each of them s a copy of N Moreover under the nherted Sefert structure of N, N1 ¼ M(0, 1Á 12, 12) and N2 ¼ M( 1, 1Á ) If we consder that M( 1, 0Á 12, 12, z) s obtaned by dentfyng N1 ¼ and N 2 ¼ along f (T ), we get a new sem-bundle structure so that f (T ) become a fber torus Snce the Euler number of the Sefert structure s z, the new glung map must be z 1 1 Ths reduces us to the stuaton that f (T ) s parallel to T ¼ Case (2) Both f (T ) N are type (II) surfaces, for 1, 2 (Fg 4) Hence f (T ) N 1 s exactly as that n Case (1) () Smlarly, f (T ) N 2 are n parallel annulus B 1,, B n located n a cyclc order n N Set B b b ¼ b 1,, b n, b1 ¼,, b¼ n are located n cyclc order n N 2 We can assume paste a 1 to b 1 and paste a 2 to b 2 (re-ndex B f needed) Then we have a s pasted to b, and a ¼ pasted to b ¼, 1,, n So A and B are pasted to one component of f (T ) n N Snce f (T ) s connected, n 1 But here f (T ) does not separate N, t s mpossble Case (3) (We copy the proof of [4] for ths case) Now = 1 z 0 1, and N M(0, 0Á 12, 12, 12, 12, z), e(n ) z (Both f (T ) N are type (III)) We may assume that f (T ) has been sotoped to be ether vertcal or horzontal n ths Sefert fberng Snce a connected horzontal essental surface s not separatng, f (T ) must be vertcal Then f (T ) must separate M(0, 0Á 12, 12, 12, 12, z) nto two copes of N both havng the nherted Sefert structure M(0, 1Á 12, 12) We can rechoose the sem-bundle structure so that f (T ) become a fber torus Then for the new torus sem-bundle structure the glung map must also be 1 z Ths reduces us to 0 1 the stuaton that f (T ) s parallel to T ¼

11 SELF-MAPPING DEGREES OF TORUS BUNDLES Incompressble surfaces n 1 fbers Lemma 25 ([4], Lemma 27) For a torus bundle M, f s not conjugate to, then any essental closed surface n M s sotopc to a unon of torus Lemma 26 If a torus sem-bundle N has no torus bundle structure, then any essental closed surface n N s sotopc to copes of torus fbers of a torus sembundle structure on N, whch s somorphc to N Proof Let F be an essental close surface n N N 1 N 2 By Lemma 22 (3), F s a unon of parallel tor For our purpose we may assume that F s a torus Isotope F so that F N s essental n N Then each component of F N must be n one of the three types lsted n Lemma 22 If F N s of type (I), then the proof s fnshed There are two cases remanng: (a) Both F N are of type (II) for 1, 2 (Fg 4) Then N Ò F are I-bundles over N F Glung those two I -bundles along N wll get an I-bundle over F and N obtaned from ths I-bundle by dentfyng ts top and bottom, whch provdes a torus bundle structure of N (b) Some F N s of type (III), say 2 (Fg 5) Then F s the same as f (T ) ether n Case (1) or Case (3) of the proof of Theorem 24, depends on F N 1 s of type (III) or type (II) As ndcated n the proof of Theorem 24, we can rechoose the new torus sembundle structure N so that F become a fber torus; moreover f choosng sutable coordnates, we can make to be 24 Coordnates of torus sem-bundles Call a map gï (M, M) (M ¼, M ¼ ) s proper f g 1 ( M ¼ ) M Lemma 27 If V T I wth the two boundares T, T and gï (V, T, T ) (N, N) s a proper map, then (g T ) (g T ), where 0 1 Proof Let pï T I N be the double coverng and be the deck transformaton map Snce g ( 1 (V )) (g T ) ( 1 (T )) 1 ( N) 1 (N), thus g can be lfted to a map ÉgÏ V T I s

12 142 H SUN, S WANG AND J WU Fg 7 N s double covered by M 1 From the commuted dagram above, we have: g T p T 1 Ég T, g T p T 1 Æ T 0 Æ Ég T d We can choose coordnate on (T I, T 0, T 1), such that p T 1 When consderng fundamental group, we have (Ég T ) (Ég T ) Thus by the above equaton: (g T ) (g T ) where ( T 0) 0 1 Lemma 28 A torus sem-bundle N s doubly covered by a torus bundle M 1 where (x, y) (x, y) wth sutable choce of coordnate (x, y) on the torus Proof Let N N 1 N 2 wth N 1 N 2 T Let pï M N be the double cover, where M s a torus bundle, p 1 (N ) M s homeomorphc to T I, p 1 (T ) T 1 T 2 Cut M along T 1, T 2, get M Ò T 1 T 2 The two boundares of M are denoted by T and T ¼, T 1 s pasted to T 2 by, T1 ¼ of these are shown n Fg 7 s pasted to T ¼ 2 by ¼ Let p p M All We can choose coordnate on T 1, T 2, such that (p T ) d Snce T ¼ to T, we can dentfy 1 (T ¼) wth 1(T ) By Lemma 27, we have (p T ¼ (p T ) s parallel )

13 SELF-MAPPING DEGREES OF TORUS BUNDLES 143 From Fg 7, we know that (p2 T2 ) Æ (p 1 T1 ), (p 2 T ¼ 2 ) Æ ¼ Æ (p 1 T ¼ 1 ) Then we get, ¼ Æ Æ Thus M has the torus bundle structure M ¼ 1 M z z 1 By Theorem 24, and the fact that, wth sutable choce of canoncal coordnates of N, we can set s one of the four matrces: 0 1, 1 z 1 z, and a b where abcd 0, ad bc z 1 c d When s n the frst three matrces, N s a Sefert manfold wth Euler number z N s E 3 manfold f z 0 and s Nl manfold f z 0 Now suppose a b where abcd 0, ad bc 1 Then by Lemma 28, N s double covered by M 1 Snce we have ( 1 ) 1 ad bc 2ab 2cd ad bc Trace(( 1 ) ) 2ad bc 2ad bc2bc 22bc1 2 By Proposton 13, M 1 admts Sol geometry, thus N admts Sol geometry The frst part of Proposton 15 s proved If N also has torus bundle structure, t must have non-separatng essental torus Recall the proof of Lemma 26, an essental torus n N can be non-separatng only f case (a) s happened, and n ths case, c d under sutable choce of canoncal z 1 coordnates, and N does have torus bundle structure Ths fnshes the moreover part of Proposton Lftng automorphsm from sem-bundle to bundle Theorem 29 Suppose f Ï N N s a non-zero degree map and f 1 (T ¼ ) s a unon of copes of T, where T, T ¼ are the torus fber of N, N respectvely Then we

14 144 H SUN, S WANG AND J WU have commute dagram where M, M ¼ are the torus bundle whch are double covers of N, N and É f Ï M M ¼ s a lft of f respectvely Proof We only have to check f (p ( 1 (M))) p ¼ ( 1(M ¼ )) Let É T, É T ¼ be one of the lftng of T, T ¼ n M, M ¼ respectvely In torus bundle M, we have the exact sequence: 1 1 (É T ) 1 (M) 1 (S 1 ) 1 In torus sem-bundle N, we have another exact sequence: 1 1 (T ) 1 (N ) Snce f 1 (T ¼ ) s a unon of copes of T, we can assume f (T ) T ¼ Then we have the commuted dagram (every row s exact): here Æp, Æp ¼, Æ f are the maps among the fundamental groups of the base spaces of fber bundles nduced by the maps among the fundamental groups of the total spaces We present the group 2 2 by a, b a 2 b 2 1 and choose the generator a, b such that Æp (1) ab, Æp ¼ (1) ab (here 1 s the generator of 1(S 1 )) Snce a 2 b 2 1, so Æ f (a) 2 Æ f (b) 2 1, then Æ f (a), Æ f (b) must be of the form ab ba or ba ab, and Æ f (ab) (ab) k or (ba) k (ab) k So Æ f (Æp ( 1 (S 1 ))) Æp ¼ ( 1(S 1 ))

15 SELF-MAPPING DEGREES OF TORUS BUNDLES 145 For any «¾ 1 (M), let f (p («)) Snce j 2 ( ) Æ f (Æp (É j1 («))) ¾ Æp ¼ ( 1(S 1 )), and there s ¾ 1 (M ¼ ) such that Æp ¼ ( É j2 ( )) j 2 ( ), so j 2 (p ¼ ( ) 1 ) Æp ¼ ( É j2 ( )) j 2 ( 1 ) j 2 ( ) j 2 ( 1 ) 1 Snce (p ¼ ) s an somorphsm, there s Æ ¾ 1 (É T ¼ ) such that 2 ((p ¼ ) (Æ)) p ¼ ( ) 1 We have p É ¼ ( 2 (Æ 1 ) ) 2 ((p ¼ ) (Æ )) 1 p ¼ ( ) (p¼ ( ) 1 ) 1 p ¼ ( ) So f (p ( 1 (M))) p ¼ ( 1(M ¼ )), thus É f exsts 3 The degrees of self maps of torus bundles We are gong to prove Theorem 16 (ref Proposton 13) There are two cases to consder: CASE 1: s conjugated to Now M n 1 s a Sefert manfold whose Euler number of Sefert fberng e(m ) s equal to n (1I) If n 0, M s T 3 or S É 1 S É 1 S 1 Here, any 2 2 nteger 0 1 matrx A commutes wth, so M admts self maps of any degrees (1II) If n 0, for a none zero degree map fï M M, by [12, Corollary 04], f s homotopc to a coverng map gï M M We can choose a sutable Sefert fberng of M such that g s a fber preservng map Denote the orbfold of M by O(M ) By [10, Lemma 35], we have: (31) m, e(m l ) e(m ) deg(g) l m, where l s the coverng degree of O(M ) O(M ) and m s the fber degree Snce e(m ) 0, from equaton (31) we get l m Thus deg( f ) deg(g) s a square number Conversely, gven a square number l 2, t s easy to construct a coverng map f Ï M M of degree l 2 CASE 2: a b c d s not conjugated to n 1 Theorem 31 Suppose s not conjugated to n 1 M admts a self map of degree l 0 f and only f there exst a 2 2 nondegenerate nteger matrx A and a postve nteger k such that l k det(a) and A ( ) k A where 1 Proof For a torus fber T ¾ M, T s ncompressble Suppose f Ï M M s a self-map of degree l 0 By [6, Lemma 65], f s homotopc to gï M M

16 146 H SUN, S WANG AND J WU Fg 8 Non-zero degree self-map of M such that g 1 (T ) s an ncompressble surface of M Thus by Lemma 25, g 1 (T ) s sotopc to a unon of torus fbers Suppose M Òg 1 (T ) has k components V 1,, V k Each V s a T I Denote two torus boundary components of V by T and T, and the homeomorphsm glung T to T 1 by see Fg 8 Then M k Æ Æ 1 M By choosng sutable coordnate on the torus fber, we have k Æ Æ 0, 1 accordng to Theorem 23 Below we assume k Æ Æ 0 (replace by 1 f needed) Let ÉgÏ M Ò g 1 (T ) M Ò T be the map nduced by g We have the followng commuted dagram: (32) Denote the restrcton of Ég to V by g From the commuted dagram n Fg 8, we have: (33) g 1 T 1 Æ Æ g T, where 1, 1,, k and f k then 1 s 1

17 Snce T (g T SELF-MAPPING DEGREES OF TORUS BUNDLES 147 s parallel to T, we can dentfy 1 (T ) wth 1 (T ) Thus (g T ) ) and ( k ) ( 1 ) on fundamental group The dentty (33) deduces that: Set A (g 1 T 1 ) and get: (g 1 T 1 ) (g 1 T 1 ) ( k ) ( 1 ) (g k1 T k1 ) ( k ) ( 1 ) (g k T k ) ( k 1 ) ( 1 ) (g k T k ) ( k 1 ) ( 1 ) ( ) k (g 1 T 1 ) (34) A ( ) k A Clearly deg(g) kdet(a) The sgn of deg(g) s decded by and the sgn of det(a) Thus l deg( f ) deg(g) k det(a) Conversely, we set 1 k 1 d, k and construct the map ÉgÏ M Ò g 1 (T ) M ÒT such that Ég V Æ A) dï T ( ( 1) I T I for 1,, k Ths constructon fts the commuted dagram (32) Thus we get the quotent gï M M whose degree s equal to k det(a) Suppose A p q r s where p, q, r, s ¾ We use equaton (34) to solve p, q, r, s and then can determne l by Theorem 31 (2I) If s Anosov whch means the absolute value of one egenvalue of s larger than 1 whle the other s less than 1 In ths case, the k n the equaton (34) must be equal to 1 We have: p q a b a b p q Solve ths matrx equaton and get: r s A ¼ ¼ p r c d c br c cp (d a)r c p(d a) br c p r p where brc, (d a)rc, (p(d a) br)c ¾ d ½ r ( 1), s ½ ( 1)

18 148 H SUN, S WANG AND J WU By Theorem 31, we have: (2II) l p 2 (d a) c pr b c r 2 If s perodc, may assume s ether (A) If , or ( has order 3), the equaton (34) means: A A (k 0 mod 3), A (k 1 mod 3), A (k 2 mod 3) 2 After solvng all the above possble cases, we get: p q (k 1 mod 3, 1), q p q p q (k 1 mod 3, 1), q p p A p q (k 2 mod 3, 1), q p p p q (k 2 mod 3, 1) q p q, whch nduces degree 0 map 0 0 By Theorem 31: If k 0 mod 3, we have A 0 0 It s easy to deduce that: l k (p 2 pq q 2 ) (k 1 mod 3), k ( p 2 pq q 2 ) (k 2 mod 3) l (3t 1)(p 2 pq q 2 ), t, p, q ¾ The same method s appled to the other two cases and we get:, then: (B) If 0 1 l (4t 1)(p 2 q 2 ), t, p, q ¾ 0 1, or

19 (C) If SELF-MAPPING DEGREES OF TORUS BUNDLES 149, then: l (6t 1)(p 2 pq q 2 ), t, p, q ¾ 4 The degrees of self maps of torus sem-bundles We are gong to prove Theorem 17 (ref Proposton 15) We wll assume that torus sem-bundle N consdered n ths secton has no torus bundle structure, otherwse D(N ) s determned n Secton 3 s l 0 and T s a torus fber of N By Suppose the degree of f Ï N N [6, Lemma 65], f s homotopc to gï N N such that g 1 (T ) s ncompressble n N Thus by Lemma 26 and ts proof (also ref the proof of Theorem 24), we have g 1 (T ) s sotopc to ether a unon of torus fbers, or a unon of torus fbers of another sem-bundle structure whch s somorphc to the orgnal one Also the later case happen only f N s a Nl manfold Note by Theorem 29 and the proof n Secton 3 (1II), Nl 3-manfolds admts no orentaton reversng homeomorphsm Suppose now g 1 (T ) has k connected components, then N Ò g 1 (T ) has two copes of N, denoted by V 0 and V k, and k 1 copes of T I, denoted by V, 1,, k 1 Denote the boundares of V 0 and V k by T 0 and T k, the boundares of V by T and T, 1,, k 1, and the glung map from T to T 1 by ( 0,, k 1) see Fg 9 Then N k 1 Æ Æ 0 N, and k 1 Æ Æ 0, 1 by Theorem 24 (wth a sutable orentaton of the canoncal coordnate) Below we assume k 1 Æ Æ 0 (replace by 1 f needed) Let ÉgÏ N Ò g 1 (T ) N Ò T be the map nduced by g, and we have commuted dagram: (41) Snce T s parallel to T, we can dentty 1 (T ) wth 1 (T ) ( 0,, k 1) Thus ( k 1 ) ( 0 ) on fundamental group Denote the restrcton of Ég on V by g Then gï V N 1 f even, and gï V N 2 f odd Lemma 41 Under the canoncal bass (l 0, l ½ ), (g 0 T 0 ) s of the form 2m 0 n where n 0, m, n ¾, and so s (g k T k ) Proof Let gï N N be a proper map, we argue that under the bass (l 0, l ½ ), (g N ) s of the form 2m 0 n where n 0, m, n ¾

20 150 H SUN, S WANG AND J WU Fg 9 Non-zero degree self-map of N Choose a presentaton 1 (N)a, b a bab wth l 0 a 2 and l ½ b Suppose g (a) a m¼ b q, g (b) a p b n Snce g (a) g (b)g (a)g (b), we get: a m¼ b q a p b n a m¼ b q a p b n a m¼ 2p b ( 1)m¼ p n( 1) p qn Thus: m ¼ m ¼ 2p, q ( 1) m¼ p n ( 1) p q n, Àµ p 0, m ¼ odd, or p 0, n 0 Abandon the case that p n 0 for g 0 s non-zero degree map and let m ¼ 2m 1, we get: g (a) a 2m1 b q, g (b) b n Snce 1 ( N)a 2, b [a 2, b]1 and g (a 2 )a 2m1 b q a 2m1 b q a 4m2, we have (g N ) 2m 0 n Theorem 42 If N has no torus bundle structure, then N admts a self map of degree l 0 f and only f there exst a postve nteger k and two nteger matrces A 1, A 2 of form 2m, m, n ¾, n 0, satsfyng the followng equaton: 0 n A 2 ( ) s 1 A 1 (k 2s), ( ) s A 1 (k 2s 1), such that l k det(a 1 ) where 1

21 SELF-MAPPING DEGREES OF TORUS BUNDLES 151 Proof From Fg 9, we know that: (42) g 1 T where 1, 0,, k 1 Thus f k 2s s even, then: (43) Æ 1 Æ g T Æ g T ( 0 mod 2), ( 1 mod 2), (g k T ) k (g k T ) ( k k 1 ) ( 0 ) by Fg 9 If k 2s 1 s odd, then: (g k 1 T k 1 ) ( k 2 ) ( 0 ) by (42) (g k 1 T k 1 ) ( k 2 ) ( 0 ) by Lemma 28 ( ) s 1 (g 0 T 0 ) (44) (g k T ) k (g k T ) ( k k 1 ) ( 0 ) (g k 1 T k 1 ) ( k 2 ) ( 0 ) (g k 1 T k 1 ) ( k 2 ) ( 0 ) ( ) s (g 0 T 0 ) It s easy to see that deg(g) kdet(g 0 T 0 ) The sgn of deg(g) s decded by both and the sgn of det(g 0 T 0 ) Thus l deg( f ) deg(g) k det(g 0 T 0 ) Fnally by applyng Lemma 41, we fnsh the proof of one drecton of Theorem 42 Conversely, f gven k, A 1, A 2, then we can easly construct the maps g 0, g k Ï N N such that (g 0 T 0 ) A 1, (g k T ) A k 2 Set 0 k 2 d, k 1 and g Ï T I N ( 1,, k 1) s a map such that: g T Æ g 1 T Æ g 1 T 1 ( 1 mod 2), 1 ( 0 mod 2) Then Ég Ë g fts the commutatve dagram (41) Thus we get the quotent map gï N N of degree k det(a 1 ) Gven a b ¾ GL2 () and suppose (g 0 T 2m ¼ 0 n ¼ c d where m, n, m ¼, n ¼ ¾ 0 ) 2m 0 n, (g k T k )

22 152 H SUN, S WANG AND J WU CASE 1: abcd 0, ad bc 1 (It should be noted that ( 1 ) s Anosov) Snce gï N N satsfes g 1 (T ) s copes of torus fber, by Theorem 29 g can be lft to g Ï ¼ M 1 M 1 By the argument of Anosov monodromy case n Secton 3, the degree of g ¼ n the S 1 drecton s 1 So we have k 1 By equaton (44), we have: (g 1 T 1 ) (g 0 T 0 ) If 1, then: 2m ¼ 0 n a b ¼ c d a c b d 2m 0 n Solvng ths matrx equaton we have: Thus (g 0 T 0 ) 2m 0 2m 1 n 2m 1, m ¼ m, n ¼ 2m 1 whch means: deg(g) k det((g 0 T 0 ) ) (2m 1) 2 If 1, then: 2m ¼ 0 n a b ¼ c d a c b d 1 2m 0 n Solvng ths matrx equaton we have: n (2m ¼ 1), (2m ¼ 1) a (2m 1) d, n ¼ (2m 1) Suppose (2m 1) u agcd(a, d), then both u and agcd(a, d) must be odd Smlarly, snce n 2m ¼ 1 u dgcd(a, d) s odd, then dgcd(a, d) s odd also Thus (g 0 T 0 ) u agcd(a, d) 0 0 u dgcd(a, d) whch means: deg(g) k det((g 0 T 0 ) ) u 2 ad gcd(a, d) 2 Ths degree can be realzed here f and only f adgcd(a, d) 2 s odd

23 SELF-MAPPING DEGREES OF TORUS BUNDLES 153 CASE 2: abcd 0 Then there are three subcases (2I) z 1 In ths case N s a torus bundle whch has been dscussed n Secton 3 (2II) 0 1, or equvalently z 1 1 z When z 0, we dscuss the followng four possble cases: (A) equaton: If 1 and k 2s s even, then by equaton (43), we have the followng 2m ¼ 0 n 0 1 ¼ 1 z ( 1) s 1 zk 0 1 Ths equaton has no soluton (B) If 1 and k 2s s even, then by equaton (43): 2m ¼ 0 n 0 1 ¼ 1 z ( 1) s zk 1 2m 0 n 2m 0 n Ths equaton has no soluton ether (C) If 1 and k 2s 1 s odd, then by equaton (44): 2m ¼ 0 n 0 1 ¼ 1 z Solvng ths matrx equaton: ( 1) s kz n ( 1) s (2m ¼ 1), n ¼ ( 1) s (2m 1), n ¼ ( 1) s kn So 2m 1 kn, thus k s odd, f k exsts Then (g k T ) 2m ¼ k 0 k(2m ¼ 1) whch means: 2m 0 n deg(g) k det((g 0 T 0 ) ) k det((g k T k ) ) k 2 (2m ¼ 1) 2 Ths degree s an odd square number In another hand, when k 1, all odd square number can be realzed as a degree: (g k T ) 2m ¼ k 0 2m ¼ 1 (D) If 1 and k 2s 1 s odd, then by equaton (44): 2m ¼ 0 n 0 1 ¼ 1 z ( 1) s zk 1 Ths equaton has no soluton 2m 0 n

24 154 H SUN, S WANG AND J WU When z 0, the same method wll show that deg(g) s odd, and all odd numbers 0 1 In ths case, deg(g) can be determned as n case (2II) can be realzed (2III) 1 z ACKNOWLEDGEMENTS The paper s enhanced by the referee s comments The authors are supported by grant No of the Natonal Natural Scence Foundaton of Chna References [1] XM Du: On self-mappng degrees of S 3 -geometry 3-manfolds, to appear n Acta Math Sn (Engl Ser) [2] H Duan and S Wang: The degrees of maps between manfolds, Math Z 244 (2003), [3] E Ghys and V Sergescu: Stablté et conjugason dfférentable pour certans feulletages, Topology 19 (1980), [4] A Hatcher: Notes on basc 3-manfold topology, ~hatcher/ [5] C Hayat-Legrand, E Kudryavtseva, SC Wang and H Zeschang: Degrees of self-mappngs of Sefert manfolds wth fnte fundamental groups, Rend Istt Mat Unv Treste 32 (2001), [6] J Hempel: 3-Manfolds, Prnceton Unv Press, Prnceton, NJ, 1976 [7] K Ireland and M Rosen: A Classcal Introducton to Modern Number Theory, second edton, Graduate Texts n Mathematcs 84, Sprnger, New York, 1990 [8] SV Matveev and AA Perfl ev: Perodcty of degrees of mappngs between Sefert manfolds, Dokl Akad Nauk 395 (2004), [9] M Sakuma: Involutons on torus bundles over S 1, Osaka J Math 22 (1985), [10] P Scott: The geometres of 3-manfolds, Bull London Math Soc 15 (1983), [11] WP Thurston: Three-dmensonal manfolds, Klenan groups and hyperbolc geometry, Bull Amer Math Soc (NS) 6 (1982), [12] SC Wang: The 1 -njectvty of self-maps of nonzero degree on 3-manfolds, Math Ann 297 (1993),

25 SELF-MAPPING DEGREES OF TORUS BUNDLES 155 Hongbn Sun School of Mathematcal Scences Pekng Unversty Bejng Chna e-mal: Shcheng Wang School of Mathematcal Scences Pekng Unversty Bejng Chna e-mal: Janchun Wu School of Mathematcal Scences Pekng Unversty Bejng Chna e-mal: