TitleON THE S^1-FIBRED NILBOTT TOWER. Citation Osaka Journal of Mathematics. 51(1)

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1 TitleON THE S^-FIBRED NILBOTT TOWER Author(s) Nakayama, Mayumi Citation Osaka Journal of Mathematics. 5() Issue Date Text Version publisher URL DOI Rights Osaka University

2 Nakayama, M. Osaka J. Math. 5 (204), ON THE S -FIBRED NILBOTT TOWER MAYUMI NAKAYAMA (Received January 6, 202, revised May, 202) Abstract We shall introduce a notion of S -fibred nilbott tower. It is an iterated S -bundle whose top space is called an S -fibred nilbott manifold and the S -bundle of each stage realizes a Seifert construction. The S -fibred nilbott tower is a generalization of real Bott tower from the viewpoint of fibration. In this note we shall prove that any S -fibred nilbott manifold is diffeomorphic to an infranilmanifold. According to the group extension of each stage, there are two classes of S -fibred nilbott manifolds which is defined as finite type or infinite type. We discuss their properties. Contents. Introduction Seifert construction S -fibred nilbott tower Proof of Theorem Torus actions on S -fibred nilbott manifolds dimensional S -fibred nilbott towers dimensional S -fibred nilbott manifolds of finite type dimensional S -fibred nilbott manifolds of infinite type Realization Realization of S -fibration over a Klein bottle K Realization of S -fibration over T Halperin Carlsson conjecture References Introduction Let M be a closed aspherical manifold which is the top space of an iterated S -bundle over a point: (.) M M n M n M {pt}. Suppose X is the universal covering of M and each X i is the universal covering of M i and put (M i ) i (i,, n ) and (M) Mathematics Subject Classification. 53C55, 57S25, 5M0.

3 68 M. NAKAYAMA DEFINITION.. An S -fibred nilbott tower is a sequence (.) which satisfies I, II and III below. The top space M is said to be an S -fibred nilbott manifold (of depth n). I. Each M i is a fiber space over M i with fiber S. II. For the group extension (.2) i i associated to the fiber space I, there is an equivariant principal bundle: (.3) Ê X i p i Xi. III. Each i normalizes Ê. The purpose of this paper is to prove the following results. Theorem.2. Suppose that M is an S -fibred nilbott manifold. (i) If every cocycle of H 2( i, ) which represents a group extension (.2) is of finite order, then M is diffeomorphic to a Riemannian flat manifold. (ii) If there exists a cocycle of H 2( i, ) which represents a group extension (.2) is of infinite order, then M is diffeomorphic to an infranilmanifold. In addition, M cannot be diffeomorphic to any Riemannian flat manifold. As a consequence, we have the following classification. (See Proposition 4. and Proposition 4.2.) Proposition.3. The 3-dimensional S -fibred nilbott manifolds of finite type are those of G, G 2, B, B 2, B 3, B 4. Proposition.4. Any 3-dimensional S -fibred nilbott manifold of infinite type is either a Heisenberg nilmanifold N½(k) or an Heisenberg infranilmanifold N¼(k). Real Bott manifolds consist of G, G 2, B, B 3 among these G, G 2, B, B 2, B 3, B 4. (Refer to the classification of 3-dimensional Riemannian flat manifolds by Wolf [3]. We quote the notations G i, B i there.) Masuda and Lee [8] have also proved the above results. By (.2) of Definition., a 3-dimensional S -fibred nilbott manifold M gives a group extension: (M) Q where Q is the fundamental group of a Klein Bottle K or a torus T 2. Then this group extension gives a 2-cocycle in the group cohomology H 2 (Q, ) with a homomorphism Ï Q Aut() { }. Conversely we have shown

4 ON THE S -FIBRED NILBOTT TOWER 69 Theorem.5. Every cocycle of H 2 (Q, ) can be realized as a diffeomorphism class of an S -fibred nilbott manifold. 2. Seifert construction We shall explain the Seifert construction briefly. It is a tool to construct a closed aspherical manifold for a given extension. Let (2.) ½ Q be a group extension and Ï Q Aut(½) a conjugation function defined by a section s Ï Q for the projection. Define f Ï Q Q ½ by s(«)s( ) f («, )s(«). Then f defines the group which is the product ½ Q with the group law: (2.2) (n, «)(m, ) (n («)(m) f («, ), «). (n, m ¾ ½, «, ¾ Q) (cf. [0] for example). Suppose ½ is a torsionfree finitely generated nilpotent group. By the Mal cev s unique existence theorem, there is a simply connected nilpotent Lie group N containing ½ as a discrete uniform subgroup. (See [2] for example.) Moreover if Q acts smoothly and properly discontinuously on a contractible smooth manifold W such that the quotient space WQ is compact, then there is a map Ï Q Map(W, N ) whose images consist of smooth maps of W into N satisfying: (2.3) f («, ) ( Æ(«) Æ ( ) Æ «) («) («) («, ¾ Q), here ÆÏ Q Aut(N ) is the unique extension of by Mal cev s unique existence property. We simply write f Æ for (2.3). And an action of on N W is obtained by (2.4) (n, «)(x, Û) (n Æ(«)(x) («)(«Û), «Û). This action (, N W ) is said to be a Seifert construction. (See [5] for details.) In particular, when Q is a finite group F and W {pt} it follows Map(W,N ) N for which there is a smooth map Ï F N satisfying f Æ : (2.5) f («, ) Æ(«)(( )) («) («) («, ¾ F). Let E(N ) be a semidirect product N K with K be a maximal compact subgroup of Aut(N ). And we can define a discrete faithful representation Ï E(N ) by (2.6) ((n, «)) (n («), ((«) ) Æ Æ(«)), (here is a conjugation map). Then the action of on N is defined by (2.7) (n, «)(x) ((n, «))(x) n Æ(«)(x) («).

5 70 M. NAKAYAMA Note that the action (,N ) is a Seifert construction and if is torsionfree N is an infranilmanifold (cf. [5] or [0]). 3. S -fibred nilbott tower In this section we shall gives a proof of Theorem.2 of Introduction and apply our theorem to torus actions. 3.. Proof of Theorem.2. Suppose that (3.) M M n S M n S S M S {pt} is an S -fibred nilbott tower. fiber space; By the definition, there is a group extension of the (3.2) i i for any i. The conjugate by each element of i defines a homomorphism Ï i Aut() { }. With this action, is a i -module so that the group cohomology H ( i, ) is defined. Then the above group extension (3.2) represents a 2-cocycle in H 2 ( i, ) (cf. [0]). Proof of Theorem.2. Given a group extension (3.2), we suppose by induction that there exists a torsionfree finitely generated nilpotent normal subgroup ½ i of finite index in i such that the induced extension ɽ i is a central extension: (3.3) i i ɽ i ½ i. It is easy to see that ɽ i is a torsionfree finitely generated normal nilpotent subgroup of finite index in i. Then i is a virtually nilpotent subgroup, i.e. ɽ i i F i where F i i ɽ i is a finite group. Let Ni É, N i be a nilpotent Lie group containing ɽ i, ½ i as a discrete cocompact subgroup respectively. Let A( Ni É ) Ni É Aut( Ni É ) be the affine group. If Ki É is a maximal compact subgroup of Aut( Ni É ), then the subgroup E( Ni É ) Ni É Ki É is called the euclidean group of Ni É. Then there exists a faithful homomorphism (see (2.6)): (3.4) i Ï i E( É Ni ) for which i ɽi id and the quotient É Ni i ( i ) is an infranilmanifold. The explicit

6 formula is given by the following ON THE S -FIBRED NILBOTT TOWER 7 (3.5) i ((n, «)) (n («), ((«) ) Æ Æ(«)) for n ¾ ɽ i, «¾ F i where Ï F i Ni É, Æ Ï F i Aut( Ni É ). As ɽ i Ni É, there is a -dimensional vector space R containing as a discrete uniform subgroup which has a central group extension (cf. [2]): R É Ni N i where N i Ni É R is a simply connected nilpotent Lie group. As R ɽ i is discrete cocompact in R and R ɽ i ɽ i ½ i is an inclusion, noting that ½ i is torsionfree, it follows that R ɽ i. We obtain the commutative diagram in which the vertical maps are inclusions: ɽ i ½ i (3.6) R É Ni N i. On the other hand, (3.4) induces the following group extension: (3.7) i p i i Ç i i i ( i ) Ç i ( i ). Since ɽ i and É Ni centralizes and R respectively, Ç i is a homomorphism from i into E(N i ). The explicit formula is given by the following: (3.8) Ç i ((Æn, «)) (Æn Æ(«), ( Æ(«) ) Æ Ç(«)) for Æn ¾ ½ i, «¾ F i where Æ p i Æ Ï F i N i, Ç Ï F i Aut(N i ); Ç(«)( Æx) Æ(«)(x). Using (.3) and Mal cev s unique extension property (compare [2]), it is easy to check that the above Ç Ï F i Aut(N i ) is a well-defined homomorphism. Thus we obtain an equivariant fibration: (3.9) (, R) ( i ( i ), Ni É ) i ( Ç i ( i ), N i ).

7 i 72 M. NAKAYAMA Suppose by induction that ( i, X i ) is equivariantly diffeomorphic to the infranilaction ( Ç i ( i ), N i ) as above. We have two Seifert fibrations from (.3): and (3.9): (, R) ( i, X i ) p i ( i, X i ) (, R) ( i ( i ), Ni É ) i ( Ç i ( i ), N i ). As i Ï i i ( i ) is isomorphic such that i id, the Seifert rigidity implies that ( i, X i ) is equivariantly diffeomorphic to ( i ( i ), Ni É ). This shows the induction step. If M X, then (, X) is equivariantly diffeomorphic to an infranil-action ((), N) É for which Ï E( N) É is a faithful representation. We have shown that M is diffeomorphic to an infranilmanifold N(). É According to Cases I, II (stated in Theorem.2), we prove that N É is isomorphic to a vector space for Case I or N É is a nilpotent Lie group but not a vector space for Case II respectively. CASE I. As every cocycle of H 2( i, ) representing a group extension (3.2) is finite, the cocycle in H 2 (½ i, ) for the induced extension of (3.3) that ɽ i ½ i is also finite. By induction, suppose that ½ i is isomorphic to a free abelian group i. Then the cocycle in H 2 ( i, ) is zero, so ɽ i is isomorphic to a free abelian group i. Hence the nilpotent Lie group N i is isomorphic to the vector space Ê i. This shows the induction step. In particular, i is isomorphic to a Bieberbach group i ( i ) E(Ê i ). As a consequence X is diffeomorphic to a Riemannian flat manifold Ê n (). CASE II. Suppose that i is virtually free abelian until i and the cocycle [ f ] ¾ H 2( i, ) representing a group extension i i is of infinite order in H 2 ( i, ). Note that i contains a torsionfree normal free abelian subgroup i. As in (3.3), there is a central group extension of ɽ i : (3.0) i i ɽ i i where [ i Ï i ] ½. Recall that there is a transfer homomorphism Ï H 2 ( i,) H 2 ( i, ) such that Æ i [ i Ï i ]Ï H 2 ( i, ) H 2 ( i, ), see [, (9.5) Proposition p. 82] for example. The restriction i [ f ] gives the bottom extension sequence of (3.0). If i [ f ] 0 ¾ H 2 ( 2, ), then 0 Æ i [ f ] [ i Ï i ][ f ] ¾ H 2 ( i, ). So i [ f ] 0. Therefore ɽ i (respectively É Ni ) is not abelian (respectively not isomorphic to a vector space). As a consequence, É N is a simply connected (non-abelian) nilpotent Lie group.

8 ON THE S -FIBRED NILBOTT TOWER 73 In order to study S -fibred nilbott manifolds further, we introduce the following definition: DEFINITION 3.. If an S -fibred nilbott manifold M satisfies Case I (respectively Case II) of Theorem.2, then M is said to be an S -fibred nilbott manifold of finite type (respectively of infinite type). Apparently there is no inter between finite type and infinite type. And S -fibred nilbott manifolds are of finite type until dimension 2. REMARK 3.2. Let M be an S -fibred nilbott manifold of finite type, then () is a Bieberbach group (cf. Theorem.2). By the Bieberbach Theorem, () satisfies a group extension (3.) n () H where n () Ê n, and H is the holonomy group of (). We may identify () with whenever is torsionfree. Proposition 3.3. Suppose M is an S -fibred nilbott manifold of finite type. Then the holonomy group of is isomorphic to the power of cyclic group of order two ( 2 ) s in O(n) (0 s n). Proof. Let M be an S -fibred nilbott manifold of finite type. Recall an equivariant fibration: (, R) ( i, Ni É ) p i ( i, N i ). If f is a cocycle in H 2 ( i, ) for Case I representing (3.2), then there exists a map Ï i R such that (3.2) f («, ) Æ(«)(( )) («) («) («, ¾ i ) (see [3]). Moreover let (n, «) ¾ i is given by and (x, Û) ¾ É Ni R N i, then the action of i (3.3) (n, «)(x, Û) (n Æ(«)(x) («), «Û) (n ¾, «¾ i ). See (2.4). As we have shown in Case I of Theorem.2, N i i is a Riemannian flat manifold Ê i i, we may assume that «Û b «A «Û (Û ¾ Ê i )

9 74 M. NAKAYAMA (b «¾ Ê i, A «¾ O(i )) in the above action of (3.3). Then the above action (3.3) has the formula: x n («) (3.4) (n, «) Û x where Û Here b «Æ («) 0 x,, 0 A «Û ¾ É Ni R Ê i Ê i. Suppose inductively that {A ««¾ i } ( 2 ) i. (3.5) ( 2 ) i ¼... ½ O(i ). Since Æ( i ) { }, the holonomy group H i of i is isomorphic to ( 2 ) s, (0 s i). This proves the induction step Torus actions on S -fibred nilbott manifolds. Given an effective T k -action on a closed aspherical manifold M, define an orbit map eú Ï T k M by eú(t) tx (x ¾ M). Then eú induces a homomorphism of the fundamental groups eú Ï (T k ) (M) which is known to be injective by Conner and Raymond [3]. But eú Ï H (T k ) H (M) is not necessarily injective. DEFINITION 3.4. When eú Ï H (T k ) H (M) is injective, we call that the T k - action is homologically injective. Corollary 3.5. Each S -fibred nilbott manifold of finite type M i admits a homologically injective T k -action where k Rank H (M i ). Moreover, the action is maximal, i.e. k Rank C( i ). Proof. We suppose by induction that there is a homologically injective maximal T k -action on M i T i H i such that k Rank H (M i ) Rank C( i ) (k 0). Since i, i are Bieberbach groups, there are two group extensions i i h i Hi, i i h i Hi where H i, H i are holonomy groups of i, i respectively and i i Ê i, i

10 i Ê i. We have a following diagram ON THE S -FIBRED NILBOTT TOWER 75 i i (3.6) i p i i h i h i H i H i Let pï Ê i R Ê i T i S T i be the canonical projection such that Ker p i i Ê i. By Proposition 3.3, H i ( 2 ) s for some s ( s i). The action ( i, Ê i ) induces an isometric action (H i, T i ) from (3.4). We may represent the action as follows: (3.7) Ç«¼ z z 2.. z i ½ ¼ t Ç«( Ç«)(z ) z 2 ¼ here Ç«h i ((n, «)) ¾ H i, t Ç«p(n («)) ¾ S, and Ï H i { } is defined by z if Æ(«), (3.8) ( Ç«)(z ) Æz if Æ(«). Note that (t Ç«) 2 p(n («))p(n («)) p(2n 2(«)). By (3.4) if Æ(«), then (3.9) (n, «) 2 x Û ¼ 2n 2(«) b «A «Û Since 2n 2(«) ¾, (t Ç«) 2 i.e. t Ç«.,. z i ¼ ¼ ½... ½½ x Û.

11 76 M. NAKAYAMA If ( Ç«) for all Ç«, it follows from (3.7) that the left translation of S on T i S T i induces an S -action on M i T i H i so that T k -action on M i T i H i follows (3.20) t t ¼ ¾ z z 2.. z i ¾ t ¼ t z ¼ z 2.. z i where (t, t ¼ ) ¾ S T k, [z,, z i ] ¾ M i T i H i. On the other hand, if there is an element Ç«of H i which ( Ç«)(z) Æz, then M i admits a T k -action by the induction hypothesis. The group extension (3.) gives rise to a group extension: ½ (3.2) [ i, i ] i [ i, i ] i i [ i, i ]. As in the proof of Proposition 3.3, [(0, «), (n, )] (((«) )(n), ). It follows that [ i, i ] {} or [ i, i ] 2 according to whether H i Ker or not. So (3.2) becomes (3.22) H (M i ) i H (M i ), or (3.23) 2 H (M i ) i H (M i ). For (3.22), it follows k Rank H (M i ) for which M i admits a homologically injective T k -action as above. For (3.23), k Rank H (M i ) and M i admits a homologically injective T k -action by the induction hypothesis. Now we show the action is maximal. Suppose ( Ç«) for all Ç«. Noting that the p i group extension i i is a central extension, we obtain a group extension: C( i ) p i p i (C( i )). On the other hand, since M i admits the above T k -action, k C( i ). Let Rank C( i ) k l, (l 0,, 2, ), then kl p i (C( i )). By the induction hypothesis, k Rank C( i ) Rank p i (C( i )). Therefore l 0 that is k Rank C( i ). Assume that there exists an element Ç«¾ H i such that ( Ç«)(z) Æz. It is easy to check that C( i ) {}, i.e. C( i ) C( i ) and since M i admits T k -action, k C( i ). By the induction hypothesis, k Rank C( i ). Hence in each case the torus action is maximal.

12 ON THE S -FIBRED NILBOTT TOWER dimensional S -fibred nilbott towers By the definition of S -fibred nilbott manifold M n, M 2 is either a torus T 2 or a Klein bottle K so that M 2 is a Riemannian flat manifold dimensional S -fibred nilbott manifolds of finite type. Any 3-dimensional S -fibred nilbott manifold M 3 of finite type is a Riemannian flat manifold. It is known that there are just 0-isomorphism classes G,,G 6, B,,B 4 of 3-dimensional Riemannian flat manifolds. (Refer to the classification of 3-dimensional Riemannian flat manifolds by Wolf [3].) In particular, for Riemannian flat 3-manifolds corresponding to B 2 and B 4, we have shown that there are two S -fibred nilbott towers: B 2 K S {pt} and B 4 K S {pt} in [0]. Remark that every real Bott manifold is an S -fibred nilbott manifold of finite type and B 2 and B 4 are not real Bott manifolds. And the following proposition has been proved. See [0] for details. Proposition 4.. The 3-dimensional S -fibred nilbott manifolds of finite type are those of G, G 2, B, B 2, B 3, B dimensional S -fibred nilbott manifolds of infinite type. Any 3-dimensional S -fibred nilbott manifold M 3 of infinite type is an infranil-heisenberg manifold. The 3-dimensional simply connected nilpotent Lie group N 3 is isomorphic to the Heisenberg Lie group N which is the product R with group law: (x, z) (y, Û) (x y Im ÆzÛ, z Û). Then a maximal compact Lie subgroup of Aut(N) is U() which acts on N (4.) e i (x, z) (x, e i z), (x, z) ( x, Æz). (e i ¾ U()), A 3-dimensional compact infranilmanifold is obtained as a quotient N¼ where ¼ is a torsionfree discrete uniform subgroup of E(N) N (U() ). See [4]. Let S M 3 M 2 be an S -fibred nilbott manifold of infinite type which has a group extension 3 2. As before this group extension contains a central group extension ɽ 3 ½ 2. Since R N is the center, this induces the commutative diagram of central extensions (cf. (3.6)): ɽ 3 ½ 2 (4.2) R N.

13 78 M. NAKAYAMA Using this, we obtain an embedding: 3 2 (4.3) R E(N) (U() ). Ç Note that (U() ) Ê 2 O(2) E(2). Since R 3 from (4.3), Ç( 2 ) is a Bieberbach group in E(2) so that Ê 2 Ç( 2 ) is either T 2 or K. Define L Ï E(N) U() to be the canonical projection. CASE (i). Suppose L( 3 ) {}. Then Ç( 2 ). So we may assume 3 ɽ 3 from (4.2). For each k ¾, we introduce the nilpotent group ½(k) which is a subgroup of N generated by c (2k, 0), a (0, k), b (0, ki). Put Z c which is a central subgroup of ½(k). It is easy to see that (4.4) [a, b] c k. Then ɽ 3 N is isomorphic to ½(k) for some k ¾. Since R is the center of N, we have a principal bundle S RZ N½(k) 2. Then the euler number of the fibration is k. (See [9] for example.) CASE (ii). Suppose that the holonomy group of 3 is nontrivial. Then we note that L( 3 ) 2 U(), but not in U(). By (3.6) L( 3 ) L( 2 ), first remark that L( 2 ) is not contained in U(). For this, suppose that (b, A) is an element of 2 Ê 2 O(2). Then for any x ¾ Ê 2, (b, A)x x, because the action of 2 on Ê 2 is free. Therefore det(a I ) 0. This implies that if A ¾ SO(2) U(), then A I. So L( 2 ) L( 3 ) is not contained in U(). Suppose that there exists an element g ¾ 3 such that L(g) (e i, ) ¾ U(). Noting (4.), it follows L(g) 2. Then L( 3 ) (U() L( 3 )) L(g). Let 3 ¼ L (U() L( 3 )) 3 which has the commutative diagram: (4.5) 3 p 3 2 ¼ 3 ¼ 2. Here 2 ¼ p 3(3 ¼). Since 2 ¼ also acts on Ê2 freely, it follows L(2 ¼) L( 3 ¼) U() L( 3 ) {}. Hence L( 2 ) L( 3 ) 2 L(g). In particular M 2 is the Klein bottle K.

14 ON THE S -FIBRED NILBOTT TOWER 79 Let n (x, 0) be a generator of N. Choose h ¾ 3 with L(h) such that the subgroup p 3 (g), p 3 (h) is the fundamental group of K. It has a relation p 3 (g)p 3 (h)p 3 (g) p 3 (h). Then n, g, h is isomorphic to 3. In particular, those generators satisfy (4.6) ghg n k h (k ¾ ), gng L(g)n n n, ahnh L(h)n n. On the other hand, fix a non-zero integer k. Let ¼(k) be a subgroup of E(N) generated by (4.7) n ((k, 0), I ), «0, k,, ((0, ki), I ), 2 where (a, x) ¾ N R E(N). Note that «2 ((0, k), I ). Then it is easily checked that (4.8) (4.9) ««n k, «n«n, n n, R E(N) (U() ) n ¼(k) Ç«, Ç. Then the subgroup generated by Ç«2, Ç is isomorphic to the subgroup of translations of Ê 2 ; t k, t Let T k 2 Ê 2 t, t 2. Then it is easy to see that the quotient [ Ç«] of order 2 acts on T 2 as (4.0) (z, z 2 ) ( z, Æz 2 ). As a consequence, Ê 2 Ç«, Ç T 2 turns out to be K. So M 3 N¼(k) is an S -fibred nilbott manifold: S N¼(k) K where S Rn is the fiber (but not an action). Compared (4.6) with ¼(k), 3 is isomorphic to ¼(k) with the following commutative arrows of isomorphisms: 3 2 (4.) n ¼(k) Ç«, Ç.

15 80 M. NAKAYAMA As both ( 3, X 3 ) and (¼(k), N) are Seifert actions, the isomorphism of (4.) implies that they are equivariantly diffeomorphic, i.e. M 3 X 3 3 N¼(k). This shows the following. Proposition 4.2. A 3-dimensional S -fibred nilbott manifold M 3 of infinite type is either a Heisenberg nilmanifold N½(k) or a Heisenberg infranilmanifold N¼(k). 5. Realization 5.. Realization of S -fibration over a Klein bottle K. Let Q be a fundamental group of a Klein Bottle K, then Q has a presentation: (5.) {g, h ghg h }. A group extension Q for any 3-dimensional S -fibred nilbott manifold over K represents a 2-cocycle in H 2 (Q, ) for some representation. Conversely, given any representation Ï Q Aut() { }, we shall prove that any element of H 2(Q, ) can be realized as an S -fibred nilbott manifold. We must consider following cases of a representation : CASE. (g), (h). CASE 2. (g), (h). CASE 3. (g), (h). CASE 4. (g), (h). Suppose i (i, 2, 3, 4) is the representation for Case i. Any element of H 2 i (Q, ) gives rise to a group extension p Q. Then is generated by Ég, É h, n such that n and p( Ég) g, p( É h) h. There exists k ¾ which satisfies (5.2) Ég É h Ég n k É h. Put i (k) for each k ¾ and [ f k ] denotes the 2-cocycle of H 2 i (Q, ) representing i(k). Note that [ f 0 ] 0. CASE : Since is trivial, H 2 (Q, ) H 2 (Q, ) H 2 (K, ) 2, and the group (k) satisfies the following presentation: (5.3) Égn Ég n, É hn É h n, Ég É h Ég n k É h. Lemma 5.. The groups (0), () are isomorphic to B, B 2 respectively.

16 ON THE S -FIBRED NILBOTT TOWER 8 Proof. First we shall discuss (0). Let Ég, h, É n ¾ (0) be as above. Put Ég, t Ég 2, t 2 n and t 3 h. É Remark that a group generated by, t, t 2, t 3 coincides with (0). Using the relation (5.3), 2 t, t 2 Ég É h Ég É h t t 3 Égn Ég n t 3. Compared these relations with those of B, (0) is isomorphic to B (due to the Wolf s notation [3]). Second, we shall discuss (). Let Ég, h,n É ¾ () be as above. Put Ég, t Ég 2, t 2 Ég 2 n and t 3 h. É A group generated by, t, t 2, t 3 coincides with (). By using the relation (5.3), 2 t, 2, t 2 Ég Ég 2 n Ég Ég n Ég Ég 2 n t, t 3 Égh Ég Ég 2 Ég 2 n É h t t 2 t This implies that () is isomorphic to B 2. (See [3].) For arbitrary k ¾, we have the following. Proposition 5.2. The group extension (k) is isomorphic to B, or B 2 in accordance with k is even or odd. 3. (5.4) Proof. Take [ f ] ¾ H 2 (Q, ) 2 by Lemma 5., then n Ég É h Ég É h (0, g)(0, h)( f (g, g), g )(0, h) f (g, h) f (g, g) f (gh, g ) f (h, h), and so (5.5) n k k f (g, h) k f (g, g) k f (gh, g ) k f (h, h). Since [k f ] ¾ H 2 (Q,), we can construct a group H k which is represented by (k f, ). Then H k is generated by the elements n and g ¼ (0, g), h ¼ (0, h) satisfying that (n, «)(m, ) (n («)(m) k f («, ), «) (n, m ¾, «, ¾ Q).

17 82 M. NAKAYAMA It follows g ¼ h ¼ g ¼ h ¼ (0, g)(0, h)( k f (g, g), g )(0, h) k f (g, h) k f (g, g) k f (gh, g ) k f (h, h) n k (from (5.5)). Thus we obtain g ¼ h ¼ g ¼ n k h ¼. In view of (5.2), a correspondence g ¼ Ég, h ¼ É h gives an isomorphism of the group extensions: H k Q (5.6) id id (k) Q. If we recall that [ f k ] (resp. [k f ]) represents (k) (resp. H k ), then it follows [ f k ] k [ f ]. Noting that [ f ] is a two torsion element, the result follows. CASE 2: Let 2 (g), 2 (h), then 2 (k) has the following presentation. (5.7) Égn Ég n, É hn É h n, Ég É h Ég n k É h, for some k ¾. Proposition 5.3. The groups 2 (0), 2 () are isomorphic to B 3, B 4 respectively. Proof. Let Ég, h, É n ¾ 2 (0) be as before. Put «h É Ég, h É, t Ég 2, t 2 h É 2 and t 3 n. Note that the group generated by «,, t, t 2, t 3 coincides with 2 (0). Using the relation (5.7), É«2 ( h É Ég) 2 h É h É Ég Ég Ég 2 t, 2 t 2, «h É h É Ég h É h É Ég t 2 «, «t 2 «É h Ég É h 2 Ég É h É h 2 t alt 3 «É h Égn Ég É h n t 3, 2, t É h Ég 2 É h É h Ég É h Ég É h É h Ég Ég Ég 2 t, t 3 É h n É h n t Since these relations correspond to those of B 3 (cf. [3]), 2 (0) is isomorphic to B 3. 3.

18 ON THE S -FIBRED NILBOTT TOWER 83 Let Ég, É h, n ¾ 2 () be as above. Put «É h Ég, n É h, t n Ég 2, t 2 É h 2, and t 3 n. Using the relation (5.7), we obtain the following presentation: É«2 ( É h Ég) 2 É hn É h Ég Ég n Ég 2 t, 2 t 2, «n É h É h Ég É hn É h Égn t 2 t 3 «, «t 2 «É h Ég É h 2 Ég É h É h 2 t «t 3 «É h Égn Ég É h n t 3, 2, t n É h n Ég 2 É hn n Ég 2 t, t 3 n É h n É hn n t 3. This implies that 2 () is isomorphic to B 4. (See [3]). Proposition 5.4. H 2 2 (Q, ) is isomorphic to 2. Proof. We first show that H 2 2 (Q, ) is a 2-torsion group. Let Q ¼ be the subgroup of Q generated by g, h 2 ¾ Q satisfying that gh 2 g (ghg ) 2 h 2. We have a commutative diagram: (5.8) 2 (k) p Q ¼ p Q ¼ where ¼ is the subgroup of 2 (k) generated by n, Ég, É h 2. Note that Ég É h 2 Ég n k É h n k É h É h 2. Since the subgroup Ég, É h2 of ¼ maps isomorphically onto Q ¼ and a restriction 2 Q ¼ id, it follows ¼ Q ¼. This shows that the restriction homomorphism Ï H 2 2 (Q,) H 2 (Q ¼, ) is the zero map, equivalently [ f k ] 0. Using the transfer homomorphism Ï H 2 (Q ¼, ) H 2 2 (Q, ) and by the property Æ ([ f ]) [Q Ï Q ¼ ][ f ] 2[ f ] ([ f ] ¾ H 2 2 (Q, )), we obtain 2[ f ] 0. Let [ f k ] be a 2-cocycle of 2 (k). Similarly as in the proof of Proposition 5.2 we obtain (5.9) [ f k ] k [ f ]. As a consequence, H 2 2 (Q, ) is isomorphic to 2.

19 84 M. NAKAYAMA The following is obvious using Proposition 5.3 and Proposition 5.4. Corollary 5.5. The group extension 2 (k) is isomorphic to B 3 or B 4 in accordance with k is even or odd. CASE 3: The group 3 (k) has the following presentation for some k ¾ ; (5.0) Égn Ég n, hn É h É n, Ég h É Ég n k h É. Lemma 5.6. (cf. (4.7).) The groups 3 (0), 3(k) are isomorphic to G 2, ¼(k) respectively. Proof. Let Ég, É h, n ¾ 3 (0) be as before. Put «Ég, t Ég 2, t 2 É h and t3 n. Note that the group generated by «, t, t 2, t 3 coincides with 3 (0). By using the relation (5.0), it is easy to check that: «2 t, «t 2 «t «t 3 «t And so 3 (0) is isomorphic to G 2. (See [3].) Suppose Ég, É h, n ¾ 3 (k) (k 0). By the relations (4.6) and (5.0), 3 (k) is isomorphic to ¼(k) (cf. (4.7)). 2, 3. Proposition 5.7. H 2 3 (G, ) is isomorphic to. Proof. From Theorem.2 and Lemma 5.6, ¼(k) represents the torsionfree element [ f k ] in H 2 3 (G, ). Moreover as in the proof of Proposition 5.2, we can show that [ f k ] k [ f ]. Therefore H 2 3 (G, ) is isomorphic to. CASE 4. The group 4 (k) has the following presentation. (5.) Égn Ég n, É hn É h n, Ég É h Ég n k É h. Put «Ég É h. It is easy to check that (5.2) «n«n, É hn É h n, «É h«n k É h. In view of (5.7), this implies that 4 (k) is isomorphic to 2 (k). We have shown that any element of H 2 i (Q, ) can be realized an S -fibred nilbott manifold M 3, and obtain the following table:

20 ON THE S -FIBRED NILBOTT TOWER 85 Case Case 2 and 4 Case 3 H 2(Q, ) 2 2 [ f ] 0 B B 3 G 2 (M 3 ) [ f ] 0: torsion B 2 B 4 [ f ]: torsionfree ¼(k) 5.2. Realization of S -fibration over T 2. Let 2 be the fundamental group of a torus T 2 generated by «,. Given a representation Ï 2 { }, we shall show that any element of H 2 (2, ) can be realized as an S -fibred nilbott manifold. We must consider following cases of a representation : CASE 5. («), ( ). CASE 6. («), ( ). CASE 7. («), ( ). Suppose i (i 5,6,7) is the representation for Case i. Any element of H 2 i ( 2,) gives rise to a group extension p 2. Then is generated by É«, É, m such that m and p( É«) «, p( É ). There exists k ¾ which satisfies (5.3) É«É É«m k É. Put i (k) for each k ¾ and [ f k ] denotes the 2-cocycle of H 2 i ( 2, ) representing i(k). Note that [ f 0 ] 0. CASE 5: The group 5 (k) has the following presentation. (5.4) É«m É«m, É m É m, É«É É«m k É, for some k ¾. Compared these relations with (4.4), Proposition 5.8. respectively. The groups 5 (0), 5 (k) are isomorphic to (T 3 ), (½( k)) Similarly as in the proof of Proposition 5.7, we obtain Proposition 5.9. H 2 5 ( 2, ) is isomorphic to. CASE 6: The group 6 (k) has the following presentation. (5.5) É«m É«m, É m É m, É«É É«m k É, for some k ¾.

21 86 M. NAKAYAMA Proposition 5.0. The groups 6 (0), 6 () are isomorphic to B, B 2 respectively. Proof. First let k 0. Put m h, É É«n, É Ég, then we can check easily that 6(0) is isomorphic to (0). So 6 (0) is isomorphic to B. Suppose k. Put m n, É«Ég, m É h, É then it is easy to check that 6 () is isomorphic to B 2. Moreover similarly as in the proof of Proposition 5.4, we obtain Proposition 5.. H 2 6 ( 2, ) is isomorphic to 2. CASE 7: The group 7 (k) has the following presentation. (5.6) É«m É«m, É m É m, É«É É«m k É, for some k ¾. Then it is easy to check that 7 (k) is isomorphic to 6 (k) if we put g É«É. We have shown that any element of H 2 (2, ) can be realized an S -fibred nilbott manifold M 3, and we obtain the following table: Case 5 Case 6 and 7 H 2 (2, ) 2 [ f ] 0 G B (M 3 ) [ f ] 0: torsion B 2 [ f ]: torsionfree ½(k) 6. Halperin Carlsson conjecture Theorem 6. (Halperin Carlsson conjecture []). Let T s be an arbitrary effective action on an m-dimensional S -fibred nilbott manifold M of finite type. Then (6.) sc j b j ( the j-th Betti number of M). In particular 2 s È m j0 Rank H j(m). Proof. By Corollary 3.5, M admits a homologically injective T k -action where k Rank C() where (M). Then we have shown in [6] that any homologically injective T k -actions on any closed aspherical manifold satisfies that kc j b j ( the j-th Betti number of M). It follows from the result of Conner Raymond [3] that there is an injective homomorphism s C(). This shows that s k so we obtain (6.2) sc j b j.

22 ON THE S -FIBRED NILBOTT TOWER 87 REMARK 6.2. Choi and Oum [2]. This result is obtained when M i is a real Bott manifold by Masuda, References [] K.S. Brown: Cohomology of Groups, Graduate Texts in Mathematics 87, Springer, New York, 982. [2] S. Choi, M. Masuda and S. Oum: Classification of real Bott manifolds and acyclic digraphs, preprint, arxiv: v, math.at (200). [3] P.E. Conner and F. Raymond: Actions of compact Lie groups on aspherical manifolds; in Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 969), Markham, Chicago, IL, 970, [4] K. Dekimpe: Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Mathematics 639, Springer, Berlin, 996. [5] Y. Kamishima, K.B. Lee and F. Raymond: The Seifert construction and its applications to infranilmanifolds, Quart. J. Math. Oxford Ser. (2) 34 (983), [6] Y. Kamishima and M. Nakayama: On the nil-bott tower of aspherical manifolds, in preparation. [7] K.B. Lee and F. Raymond: Geometric realization of group extensions by the Seifert construction; in Contributions to Group Theory, Contemp. Math. 33, Amer. Math. Soc., Providence, RI, 984, [8] J.B. Lee and M. Masuda: Topology of iterated S -bundles, preprint, arxiv: , math.at (20). [9] J. Milnor: On the 3-dimensional Brieskorn manifolds M( p, q, r); in Knots, Groups, and 3-Manifolds (Papers dedicated to the memory of R.H. Fox), Ann. of Math. Studies 84, Princeton Univ. Press, Princeton, NJ, 975, [0] M. Nakayama: Seifert construction for nilpotent groups and application to an S -fibred nil-bott Tower, to appear in RIMS Kokyuroku Bessatsu. [] V. Puppe: Multiplicative aspects of the Halperin Carlsson conjecture, Georgian Math. J. 6 (2009), [2] M.S. Raghunathan: Discrete Subgroups of Lie Groups, Ergebnisse Math. Grenzgebiete 68, Springer, New York, 972. [3] J.A. Wolf: Spaces of Constant Curvature, McGraw-Hill, New York, 967. Department of Mathematics and Information of Sciences Tokyo Metropolitan University Minami-Ohsawa -, Hachioji, Tokyo Japan nakayama-mayumi@ed.tmu.ac.jp