NilBott Tower of Aspherical Manifolds and Torus Actions

Transcription

1 NilBott Tower of Aspherical Manifolds and Torus Actions Tokyo Metropolitan University November 29, 2011 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

2 Introduction We shall explain a notion of nilbott tower to study fiber space with nil-geometry on smooth aspherical manifolds A nilbott manifold is the top space of an iterated fiber space with fiber a nilmanifold (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

3 Introduction We shall explain a notion of nilbott tower to study fiber space with nil-geometry on smooth aspherical manifolds A nilbott manifold is the top space of an iterated fiber space with fiber a nilmanifold As an application, we shall prove the smooth rigidity of nilbott manifolds of finite type Our construction applies to study homological injective toral actions on closed aspherical manifolds Then, the Halperin-Carlson conjecture is true for the homological injective T k -actions Especially, Kähler Bott manifolds are supporting examples of the Halperin-Carlson conjecture including complex Riemannian flat manifold (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

4 Definition of nilbott tower M a closed aspherical mfd associated with a tower of nil-fiber spaces: M = M n M n 1 M 1 {pt} p where each stage is a fiber bundle: L i M i i Mi 1 with fiber a nilmanifold L i = N i / i More precisely, on the universal covering M i ( i = π 1 (L i ), π i = π 1 (M i ) and so on), the following are satisfied For each i, Group extension 1 i π i π i 1 1 For each i, Equivariant principal bundle In addition, each π i normalizes N i ( i, N i ) (π i, M i ) (π i 1, M i 1 ) (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

5 Structure of nilbott manifolds From the last condition, the nilbott tower is thought of as an iterated Seifert fiber space Our advantage is to apply a rigid property of such Seifert manifolds under a rigidity condition on the base manifolds Proposition A (1) The fundamental group π i of M i is virtually polycyclic (2) If each real algebraic closure A(π i ) normalizes the nilpotent Lie group N i, then M i is diffeomorphic to the infrasolvmanifold U i /π i (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

6 Proof Using an embedding of π i into A(π i ) = U i T i U i Aut(U i ), we can construct an infrasolvmanifold U i /π i which is a fiber space over the base space again an infrasolvmanifold U i 1 /π i 1 with fiber a nilmanifold N i / i Inductively if we suppose M i 1 /π i 1 is diffeomorphic to U i 1 /π i 1, then the Seifert rigidity with the same nilfiber shows the total spaces M i /π i and U i /π i are diffeomorphic ( i, N i ) (π i, M i ) (π i 1, M i 1 ) H h ( i, N i ) (π i, U i ) (π i 1, U i 1 ) (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

7 Examples of nilbott manifolds: New & Old 1 (1) Real Bott manifolds M(A) a quotient of a flat torus T n by a free isometric action of (Z 2 ) n defined by a Bott matrix A M(A) occurs as a tower of S 1 -fiber spaces: S M(A) = M 1 S n M 1 n 1 M S1 S 1 1 {pt} π 1 (M(A)) = π E(n) = R n O(n) is a Bieberbach group M(A) = R n /π is a Riemannian flat manifold (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

8 Examples of nilbott manifolds: New & Old 2 (2) S 1 -fibred nilbott manifolds M the top space of a tower of S 1 -fiber spaces: π E(N) = N K M = M n S 1 M n 1 S 1 S1 M 1 S 1 {pt} (virtually nilpotent) M is diffeomorphic to an infranilmanifold N/π Remark that Masuda & Lee proved the similar result In this direction we introduce the following nilbott complex manifolds (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

9 Examples of nilbott manifolds: New & Old 3 (3) Holomorphic nilbott manifold M a complex manifold which is the top space of a tower of T 1 C - holomorphic fiber spaces: TC M = M 1 TC n M 1 n 1 T C M 1 TC 1 1 {pt} π 1 (M) = π is virtually nilpotent M is diffeomorphic to an infranilmanifold U/π (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

10 Finite type and Infinite type Let M be a holomorphic nilbott manifold 1 Z 2 π i π i 1 1 defines a 2-cocycle [f i ] H 2 ϕ (π i 1; Z 2 ) Definition If each [f i ] is a torsion element, then we call M a holomorphic nilbott mfd of finite type Otherwise it is infinite type Similarly, it is defined on the S 1 -fibred nilbott manifold (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

11 Theorem A If M is a holomorphic nilbott manifold of finite type, then M is diffeomorphic to a complex Riemannian flat manifold Similarly an S 1 -fibred nilbott manifold of finite type is diffeomorphic to a Riemannian flat manifold (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

12 Application 1: Homologically injective actions An effective T k -action on M the orbit map ev : T k M at x M, ev(t) = tx, the induced homomorphism ev : Z k H 1 (M; Z) Definition If ev is injective, the effective T k -action is said to be homologically injective Proposition B (Carrell(1972)) Any holomorphic (isometric) TC k -action on a closed aspherical Kähler manifold (M, Ω) is homologically injective (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

13 Halperin-Carlsson conjecture Theorem B If T k is a homologically injective action on a closed aspherical n-manifold M, then kc j b j (= the j-th Betti number of M) n In particular 2 k Rank H j (M), ie the Halperin-Carlsson conjecture is j=0 true Remark Any effective T k -action on Real Bott manifolds, Holomorphic (or S 1 -fibred ) nilbott manifolds of finite type are homologically injective (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

14 Sketch of proof 1 of Theorem B ev : Z k π 1 (M) = π induces 1 Z k π Q 1 (Q = π/z k ) Homologically injectivity = Splitting subgroup of finite index This means that There is a finite index normal splitting subgroup π π; π = Z k Q T k = R k /Z k lifts to a principal action (R k, M), M = R k W (π, M) (π, M) M/π = T k W /Q (π, M) induces (H, T k W /Q ) (here H = π/π ); α(t, z) = (t t α, αz) ( α H, (t, z) T k W /Q ) (Note that translation t α on the T k -summand) (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

15 Sketch of proof 2 of Theorem B As the homology action of H on H j (T k ) H 0 (W /Q ) is trivial H j (T k ) = H j (T k ) H 0 (W /Q ) H j (T k W /Q ) H With the aid of transfer homomorphism, the projection ν : T k W /Q T k W /Q = M/π / π/π = M H induces an isomorphism ν : H j (T k W /Q ; Q) H H j (M; Q) Hence, k C j = Rank H j (T k ) Rank H j (M; Q) = b j (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

16 Application 2: Kähler Bott tower Let TC 1 M p i i Mi 1 be a fiber space of a holomorphic nilbott tower Suppose that each M i is a Kähler manifold with Kähler form Ω i where p i : (M i, Ω i ) (M i 1, Ω i 1 ) is a Kähler map, that is preserves the Kähler form on each Moreover, if C is the lift of TC 1, then C leaves invariant a Kähler form Ω i on M i (that is, C acts as Kähler isometries wrt Ωi ) Definition The top space of the above Kähler Bott tower is called a Kähler Bott manifold (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

17 Theorem C A Kähler Bott manifold M is diffeomorphic to a complex Riemannian flat manifold TC n /F (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

18 Theorem C A Kähler Bott manifold M is diffeomorphic to a complex Riemannian flat manifold TC n /F the cocycle [f i ] for the group extension 1 Z 2 p π i i πi 1 1 has finite order M is finite type a complex Riemannian flat manifold (by Theorem A) There is an equivariant fibration: (Z 2, C) (π i, M i ) p i (π i 1, M i 1 ) Taking a finite index subgroup i in π i, there induces a central group extension: 1 Z 2 p i i i 1 1 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

19 Proof TC 1 = C/Z2, Y i = M i / i, and Y i 1 = M i 1 / i 1 (Y i as a finite covering of M i, is a Kähler manifold) a principal holomorphic fibration: TC 1 Y q i i Yi 1 The Kähler action (TC 1, Y i) is homologically injective (Proposition B) π has a finite index splitting subgroup i splits; i = Z 2 i 1 that is, H 2 ϕ ( i 1; Z 2 ) ι [f i ] = 0 Using the transfer homomorphism τ : H 2 ϕ ( i 1; Z 2 ) H ϕ (π i 1 ; Z 2 ), τ ι = [π i 1 : i 1 ] = l : H ϕ (π i 1 ; Z 2 ) H ϕ (π i 1 ; Z 2 ) As ι [f i ] = 0, l[f i ] = 0 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18

20 Final remark Although T k -actions on compact Riemannian flat manifolds are not necessarily homologically injective, we have the following result Theorem D The Halperin-Carlsson conjecture is true for any effective T k -action on compact Riemannian flat n-manifolds M Proof An effective T k -action on M induces a group extension 1 Z k π Q 1 There is a unique maximal abelian subgroup Z n by Bieberbach s Theorem Z k Z n Choosing a finite index subgroup G from Z n such that G = Z k Z n k Thus π has a finite index splitting subgroup G Apply the proof of Theorem B (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29, / 18