ALMOST FLAT MANIFOLDS WITH CYCLIC HOLONOMY ARE BOUNDARIES

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1 ALMOST FLAT MANIFOLDS WITH CYCLIC HOLONOMY ARE BOUNDARIES JAMES F. DAVIS AND FUQUAN FANG Abstract. 1. Introduction A closed manifold M is said to be almost flat if there is a sequence of metrics g i on M so that K gi diam(m, g i ) 2 0 when i, where K gi is the sectional curvature and diam(m, g i ) is the diameter of M with respect to the metric g i. Gromov [6] generalized the classical Bieberbach theorem for flat manifolds, and showed that every almost flat manifold was finitely covered by a nilmanifold, that is, the quotient of a simply-connected nilpotent Lie group by a uniform lattice. Ruh [13] strengthened Gromov s theorem and proved that an almost flat manifold is infranil, that is, diffeomorphic to a double coset space Γ\L Aut(L)/ Aut(L) where L is a simply connected nilpotent Lie group and Γ is a torsion-free subgroup of the affine group L Aut(L) so that the kernel of Γ Aut(L) has finite index in Γ and is discrete and cocompact in L. Conversely, it is not difficult to see that every infranil manifold is almost flat. Quite different from the situation for flat manifolds, there are infinitely many almost flat manifolds in every dimension greater than 2. Almost flat manifolds play a fundamental role in the study of riemannian manifolds with small volume and bounded curvature. They turn out to be the fibers of certain local fibration structures on manifolds with uniformly bounded sectional curvatures and diameters by the profound Cheeger-Fukaya-Gromov theorem [3]. Another geometric source of almost flat manifolds comes from the geometry of complete finite volume manifolds with negative sectional curvature pinched away from zero. It is well-known (see [Gr]) that any such manifold is diffeomorphic to the interior of a compact manifold whose boundary is a finite disjoint union of almost flat manifolds. This generalizes the fact that a finite volume hyperbolic manifolds is diffeomorphic to the interior of a compact manifold whose boundary is a finite disjoint union of flat manifolds. *Supported by a NSF Grant. **Supported a NSF Grant of China and the Capital Normal University. 1

2 2 JAMES F. DAVIS AND FUQUAN FANG By rescaling, one sees that for an almost flat manifold M, there is a sequence of Riemannian metrics {g i } on M so that diam(m, g i ) = 1 for all i and K gi 0 as i. By Chern-Weil theory the Pontryagin numbers of an oriented closed manifold are integrals of the Pfaffin forms on the curvature and for an almost flat manifold these integrals must converge to zero as i, since by the volume comparison theorem the sequence vol(m, g i ) is bounded above. Therefore the Pontryagin numbers of an oriented almost flat manifold all vanish since they are integers. It follows that the disjoint union of M with itself is an oriented boundary. Furthermore, if M has an almost complex structure then, by the same reasoning, all the Chern numbers of M vanish. In particular, this implies that M bounds. (Through this paper when we say M bounds we mean that M is diffeomorphic to the boundary of a compact manifold.) This clearly suggests a natural and very interesting conjecture that, almost flat manifolds are boundaries, due to Farrell and Zdravkovska [5] and S.T.Yau [16] independently. Note that by a classic theorem of Thom, a closed manifold is a boundary if and only if all its Stiefel-Whitney classes are zero. A closed oriented manifold bounds an orientable manifold if and only if all Stiefel-Whitney numbers and all Pontryagin numbers are zero. The above discussion implies that if an oriented almost flat manifold bounds, then it bounds orientably. In some special cases this conjecture had been proven. A remarkable theorem of Hamrick and Royster [8] shows that every flat manifold bounds. The holonomy of an infranil manifold is the finite group G given by the image of the fundamental group Γ in Aut(L). Farrell- Zdravkovska [5] had proved that almost flat manifolds bound provided either the holonomy group G = Z 2 or provided the holonomy group G acts effectively on the center of L. Upadhyay [15] proved that almost flat manifolds bound if all of the following conditions hold: G is cyclic, G acts trivially on the center of L, and L is 2-step nilpotent. Our main result is Theorem 1.1. Every almost flat manifold with cyclic homology group bounds. In view of Theorem 1.1, it is very interesting to ask when an almost flat (or flat) manifold bounds a compact manifold whose interior admits a finite volume complete negatively pinched (or hyperbolic) metric, as conjectured in [5]. Counterexamples were presented for flat 3-manifolds using η-invariants by Long and Reid [11]. However, it is still not known whether there is such an example of an almost flat manifold which cannot bound a negatively pinched manifold. The techniques of this paper are similar to the previous work on the problem [8], [11], and [15], in that we use the non-trivial center Z(L) of L to produce involutions of M. However we noticed that some of the

3 ALMOST FLAT MANIFOLDS 3 intermediate manifolds admit the structure of nilmanifolds, and this observation was sufficient to prove a stronger theorem. 2. almost flat manifolds with Cyclic holonomy groups Recall that a nilmanifold is a manifold diffeomorphic to the quotient of a simply connected nilpotent Lie group L by a discrete cocompact subgroup N. Thus a nilmanifold is parallelizable; indeed one projects a basis of left invariant vector fields on L to the nilmanifold. It follows that all Stiefel-Whitney numbers of a nilmanifold are zero, hence any nilmanifold bounds. An infranilmanifold is a manifold diffeomorphic to a double coset space M = Γ\L G/G where L is a simply connected nilpotent Lie group, G is a finite subgroup of Aut(L) and Γ is a discrete torsion-free cocompact subgroup of L G which maps epimorphically to G under the projection L G G. 1 Let N = Γ L. Then N is a normal subgroup of Γ and the sequence 1 N Γ G 1 is short exact. Furthermore N is a discrete cocompact subgroup of L and hence is a discrete nilpotent group. Let M = N\L G/G. The map N\L M; Nl NlG is a diffeomorphism, hence M is a nilmanifold. Left multiplication gives a Γ-action on M with quotient M and constant isotropy group N, hence M M is a regular G-cover. We call this G-action on M the affine action since every element g G lifts to an affine automorphism L L which leaves N invariant. Recall that a vector bundle E B is flat if it has finite structure group, that is, E = B H V for some finite, regular H-cover B B and some RH-module V. Such a bundle (over a CW-complex) is the pullback of the bundle EH H V along a map B BH. The tangent bundle of an infranilmanifold M with holonomy group G has flat tangent bundle T M = M G T e L. Since L is nilpotent, its center Z(L) is nontrivial. Since G is a subgroup of Aut L which leaves N invariant, it acts on L, Z(L), N, Z(N), and Z(N)\Z(L). We call these actions conjugation actions since they are related to conjugation in the group Γ. The action of Z(L) G on L G/G descends to an action of (Z(N)\Z(L)) G on M. We will show there is a finite subgroup of the torus Z(N)\Z(L) which acts on M. This action was key in all previous work on this problem [8], [5], [15] and will be for us too. To give a vague idea of how an action could be helpful, note that if there is a fixed-point free involution h : M M, 1 Note that L Aut(L) acts on L via (l, g)l = lg(l ). An equivalent definition of an infranilmanifold is a compact orbit space Γ\L for some discrete torsion-free subgroup Γ of L Aut(L) where Γ projects to a finite subgroup of Aut(L).

4 4 JAMES F. DAVIS AND FUQUAN FANG then so M bounds. ( ) M [ 1, 1] = M (m, t) (hm, t) Theorem 2.1. An infranilmanifold M = Γ\L G/G is the boundary of a compact manifold if the 2-sylow subgroup of G is cyclic. Proof of Theorem 2.1. We first reduce to the cyclic case. Let G 2 be a 2-sylow subgroup of G. Since G 2 \ M M is a odd-degree cover, the domain and range have the same Stiefel-Whitney numbers; thus one bounds if and only if the other does. Since the holonomy of G 2 \ M is G 2, in our theorem we may assume the holonomy group G is a cyclic 2-group. Let T k = Z(N)\Z(L) and let Σ be the subgroup of elements of order 2. Note Σ = Z k 2. Let Σ G be the elements of Σ invariant under the conjugation action of G. (We, following earlier authors, write Σ G instead of Σ G to avoid confusion between the fixed sets of the affine action and the conjugation action.) Since G is a finite 2-group and Σ has order a power of two, Σ G is non-trivial. Since (Z(N)\Z(L)) G acts on M and Σ G G is a subgroup, it too acts on M. This action descends to an action of Σ G on M. If the action of the elementary abelian two group Σ G has no fixed points on M, then M bounds by a result of Conner and Floyd [4, Theorem 30.1]. Hence we assume the action of Σ G on M has a fixed point x M. We also assume the conjugation action of G on Z(N) is not effective, since Proposition 1.3 of [5] states that M bounds if G acts effectively on Z(N). Remark 2.2. Suppose groups H 1 and H 2 both act freely on a set X and that the the two actions commute. Let q : X H 2 \X be the quotient map, let F be the fixed set of H 1 acting on H 2 \X, and let F = q 1 F. Then F = {x X : h 1 H 1, h 2 H 2 so that h 1 x = h 2 x}. Furthermore if x F, there is a monomorphism ϕ x : H 1 H 2 defined by the equation h 1 x = ϕ x (h 1 )x. Check This remark implies that the exponent two group Σ G is isomorphic to a subgroup of the cyclic 2-group G, and hence Σ G = Z2. We consider the case of G = Z 4 since Farrell-Zdravkovska [5] have already shown the M is a boundary when G = Z 2 and the general case of G = Z 2 k, k > 2 may be deduced in a completely similar manner. We fix a generator g of G. We will divide the proof into two cases as follows. Case (a): the conjugation action by G on Z(N)\Z(L) is trivial.

5 ALMOST FLAT MANIFOLDS 5 Note Z k 2 = Σ = Σ G = Z2, so that k = 1. We write S 1 = Z(N)\Z(L). We fix an element t S 1 of order 4. We have an S 1 G action on M and a S 1 action on M. Let M = M/ g 2. Let F be the fixed set of the action of t 2 S 1 on M. Then S 1 leaves F invariant. Let F 0 be the fixed set of t / t 2 = Z 2 on F. The following diagram is useful to keep in mind. ν 0 - F 0 F M? ˆν F 0 F M ν?? ˆν -??? ν 0 - ν - F 0 F M The four nodes in the upper left hand corner of the diagram are defined as pullbacks; hence all vertical maps are covering maps and all horizontal maps are embeddings. The notations above the horizontal arrows are not names for the inclusions, but rather they are names for normal bundles, e.g. ν is the normal bundle of F in M. (i) The following is proved in Lemmas 3.1 and 3.2 of [15]: F0, F, F 0, F 0, and F are disjoint unions of nilmanifolds, and in each case all the components have the same dimension. Furthermore, F is an infranilmanifold with holonomy cyclic of order two. (By hypothesis, M is a nilmanifold and M and M are infranilmanifolds with holonomy cyclic of order two and four respectively.) It follows that the tangent bundles of all of the above manifolds are flat. (ii) Lemma 3.3 of [15] shows that the normal bundles ν and ν 0 are flat with structure group G. More precisely, there are RG modules V and V 0 so that ν = F G V and ν 0 = F 0 G V 0 where g 2 acts by -1 on V and g acts by -1 on V 0. It follows that the G-module V and the bundle ν admit complex structures. (iii) A flat bundle with structure group of order two is either orientable, in which case all Stiefel-Whitney classes have a lift to torsion integral cohomology classes or nonorientable, in which case any Stiefel-Whitney classes is either zero or a multiple of the first Stiefel-Whitney class. This follows from the following facts: the map H (BZ 2 ; Z) H (BZ 2 ; Z 2 ) is given by Z[a]/2a Z 2 [b]; a b 2, with deg a = 2 and deg b = 1, every R[Z 2 ]-module is a direct sum of R + and R, and w 1 (EZ 2 Z2 R ) = a.

6 6 JAMES F. DAVIS AND FUQUAN FANG (iv) Farrell-Zdravkovska [5] showed that an infranil manifold M bounds if the holonomy group G is cyclic of order 2. Using (iii) we give a slight simplification of their proof. If G acts effectively on Z(L), then [5, Proposition 1.3] shows that M bounds. If M is orientable, then by (iii) any Stiefel-Whitney number of M is given by the pairing of a torsion integral cohomology class with the integral fundamental class, hence is zero, hence M bounds. If M is nonorientable, then by (iii), the only possible nonzero Stiefel-Whitney number is w1 n, [M] = 0. However, if G acts trivially on Z(L), then Proposition 2.1 of [5] shows that there is an involution on M where the components of the fixed set all have the same dimension, and hence Proposition 9.1 of Stong [14] shows w1 n, [M] = 0. Thus M bounds. (v) We now discuss flat bundles with cyclic structure group of order four. The map H (BZ 4 ; Z) H (BZ 4 ; Z 2 ) is given by Z[c]/4c Λ Z2 [d] Z 2 [e]; c e, with deg c = 2, deg d = 1, and deg e = 2. Every R[Z 4 ]-module is isomorphic to a direct sum of R +, R, and C, and w 1 (EZ 4 Z4 R ) = d, w 1 (EZ 4 Z4 C) = 0, and w 2 (EZ 4 Z4 C) = e. This last bundle has a complex structure, and hence a first Chern class, which is of order four. It follows that for any orientable flat bundle with cyclic structure group of order four, every Stiefel-Whitney class has a lift to torsion integral cohomology class. (vi) We now return to our standing assumption that M is an infranilmanifold with holonomy group G cyclic of order four, and G acts trivially on the center of L. If M is orientable, then by (v) any Stiefel-Whitney numbers is given by the pairing of a torsion integral cohomology class with the integral fundamental class, hence is zero, hence M bounds. (vii) According to an elementary argument of Conner and Floyd [4, 30.1], since M is a manifold with an involution, M is bordant to P (ν ε 1 ) = S(ν ε 1 )/x x. In particular, if [F, ν] = 0 η (BO), then M bounds. Thus it suffices to show that the Stiefel-Whitney numbers vanish. 2 w I (ν)w J (F ), [F ] 2 This leads to an exercise show that w I (ν)w J (F ), [F ] 0 implies that w K (P ), [P ] 0, where P = P (ν ɛ) and the question as to whether the first condition is stronger than the second. Here is the solution to the exercise and I believe the answer to the question is essentially no. Consider the fibration RP e i P π F and the canonical line bundle γ over P. Recall w 1 (γ) e+1 = e+1 i=1 π w i (ν)w 1 (γ) e+1 i. Note T P ɛ = π T F γ e+1. Thus it suffices to show π w I (F )π w J (ν), π ([P ] w 1 (γ) k ) for k e. But for k < e this vanishes for dimensional reasons and for k = e, π ([P ] w 1 (γ) k ) = [F ]. Check all this.

7 ALMOST FLAT MANIFOLDS 7 (viii) Since F is infranil with holonomy Z 2, its tangent bundle is flat with structure group Z 2, and hence w i (F ) is 0 or w 1 (F ) i by (iii). On the other hand, T M F and ν are flat with structure group G = Z 4, hence their difference T F is stably flat with structure group G. Thus w 1 (F ) 2 = 0 since Sq 1 H (G; Z 2 ) = 0. Hence w(f ) = 1 + w 1 (F ). Thus it suffices to show that the Stiefel-Whitney numbers and vanish. (ix) By Wu s formula w 1 (F )w I (ν), [F ] w I (ν), [F ] w 1 (F )w I (ν), [F ] = v 1 (F )w I (ν), [F ] = Sq 1 (w I (ν)), [F ] But Sq 1 w I (ν) = 0 since is a flat bundle with structure group G. Hence it suffices to show the Stiefel-Whitney numbers w I (ν), [F ] vanish. (x) Recall ν = F G V is a flat bundle with structure group G. Note that there is an involution covering the involution ˆτ : ν ν [x, y] [tx, gy] τ : F F [x] [tx] where [x] F denotes the image of x F. To see that it is well-defined we use that g and t commute (since S 1 G acts on M) and to see it is an involution we use that F is the fixed set of t 2 g 2 : M M by Remark 2.2. (xi) By Remark 2.2, F 0 = F 0 + F 0 where F ± 0 = {x M : gx = t ±1 x} Let F 0 ± ± F 0 be the images of F 0. In fact, Upadhyay [15] shows that the involution restricted to ν F + is the identity and 0 the involution restricted to ν F is multiplication by 1. 0 check

8 8 JAMES F. DAVIS AND FUQUAN FANG (xii) The diagonal involution on F [ 1, 1] (with multiplication by 1 on the interval) lifts to the diagonal involution on the bundle ν [ 1, 1]. It follows that the pair (F, ν) is bordant to the pair (P, ν) where P refers to P (ν 0 ɛ) and ν refers to the Z 2 -quotient of the bundle ν restricted to the sphere bundle. For the + components, ν is the pullback of the bundle ν F0, and hence the Stiefel-Whitney numbers w I (ν), [P ] vanish for dimensional reasons provided dim F 0 < dim F. For the components, ν = (π ν F0 ) γ and a further analysis is needed. (xiii) If dim F = dim F 0, then F is a disjoint union of nilmanifolds, hence parallelizable and orientable. To show that the Stiefel- Whitney numbers of (F, ν) vanish, we lift w I (ν) to an torsion integral cohomology class and pair it against the integral fundamental class of F to show that it vanishes. check Lemma 2.3. Let (F, ν) be an Z 2 -equivariant complex line bundle. Assume that fixed point set F 0 = F Z 2 has dimension k odd with normal bundle ν 0 isomorphic to (n k)η where η is a real line bundle, and n k odd. If w 1 (ν 0 ) 2 = 0, then w 2 (ν) k 1 2 w 1 (ν 0 )[F 0 ] = 0. We need to apply a localization result of Kosnioski and Stong to (F, ν)(cf. [KS] Proposition on page 729): Let f(x) be a polynomial of degree less than or equal to n of (n + 2)-variables, (x 1,, x n ; α 1, α 2 ) which is symmetric on the first n- variables and also symmetric on the last two variables, then f(x)[f ] = F 0 f(z, 1 + y; w 1 1, 1 + w 1 ρ; w 2 1, 1 + w 2 ρ) (1 + y) [F 0 ] where z (resp. 1 + y) corresponds to the variables from the fixed point set F 0 (resp. ν 0 ), and the last variables corresponds to variables from ν F0 in such a way that, if Z 2 is trivial on the fiber, then we omit the terms 1 + wρ 1 and 1 + wρ; 2 otherwise, if Z 2 is 1 on the fiber, we omit the terms w1 1 and wρ; 2 In our situation, let us simply take f(x) = (α 1 α 2 ) k 1 2. Since the degree of f is k 1 < n, the left side of the above equation is zero. Note that w 1 (η) 2 = 0. Hence 0 = w 2 (ν) k 1 2 w1 (η)[f 0 ] The desired result follows. The lemma implies that (F 0, ν 0, ν F0 ) bounds if dim F 0 is odd and dim F is even. Other cases are easier and follows from discussions before. This finishes the proof of the theorem in Case (a). Case (b): The conjugation action of G = Z 4 on Z(L)/Z(N) is not trivial; Recall that we have assumed that the action of G = g = Z 4 on Z(N) is not effective, hence there is a T k g 2 action on M where

9 ALMOST FLAT MANIFOLDS 9 T k = Z(N)\Z(L). Recall also that Σ is the exponent two subgroup of T k, that Σ G G acts on M, that we have assumed that Σ G has a fixed point on M, and hence Σ G = Z2. Let F be the fixed set of the Σ G -action on M and let ν be the normal bundle of F in M. As in Case (a), ν is a flat G-bundle, which restricts to 1 on g 2 G and hence ν is a complex flat bundle. Let F be the inverse image of F under the cover M M. By Remark 2.2, the Σ G action and g 2 G action on F coincide. The quotient F /Σ G is a nilmanifold again with a central action by T k /Σ G. Note that T k /Σ G G/ g 2 acts on F /Σ G. By assumption, G/ g 2 - action and T k /Σ G action does not commute, and so the conjugation representation G/ g 2 Aut(T k /Σ G ) is faithful. Thus, there is an induced invariant 2-elementary abelian subgroup Σ G/Z2 T k /Σ G as above acting on F. By Proposition 2.1 it has no fixed point. (??? maybe we know it bounds). Finally, we claim that the Σ G/ g 2 action on F lifts to a Σ G/ g 2 -action on ν. Note that the subgroup Σ G/ g 2 action on F is the reduction from an abelian group Σ G/ g 2 action on F. In fact, Σ G/ g 2 is the pre-image of Σ G/ g 2 in the following abelian extension 1 Σ G T k T k /Σ G 1 For example, if k = 1 and the holonomy ρ : G Aut(S 1 ) = g 2 is non-trivial, Σ G/Z2 = Z 4, which acts on F as a subaction of T k. Since ν = F G C k is a flat complex G-bundle, we get easily the desired lifting on ν by combining the Σ G/ g 2 -action on F and the multiplication by 1 on C k. For example, if k = 1, the involution of Σ G/ g 2 on F may be lifted to an involution on ν as follows: [x, z] [ 1x, 1z] where z C k, and 1x indicates the Σ G/ g 2 action on F. Of course here one need to verify that it is indeed an involution and the definition is well-defined. Using Stong s theorem the desired result follows. Try to give a new References [1] Baum and R.Bott, [2] G.Bredon, Introduction to Transformation Groups, [3] Cheeger, Fukaya, Gromov; JAMS 1992 [4] P.E.Conner and E.E.Floyd, Differentiable periodic maps, Springer, Berlin 1964 [5] T. Farrell and S. Zdravkovska, Do almost flat manifolds bound? Michigan Math. J., 30(1983), [6] M. Gromov, Volume and Bounded Cohomology, I.H.E.S. Pub. Math. 56(1981), [7] M. Gromov, Alomost flat manifolds, J.Differential Geom. 13 (1978), [8] G.C.Hamrick and D.C.Royster, Flat Riemannian manifolds are boundaries, Invent.Math. 66 (1982), proof of FZ, using Landweber-Stong Proposition on page 321.

10 10 JAMES F. DAVIS AND FUQUAN FANG [9] B.Lawson and Michelson, Spin Geometry, Princeton Univ. Press [10] Landweber, P.May and G. Segal [11] D. D. Long and A. W. Reid, On the geometric boundaries of hyperbolic 4- manifolds. Geom. Topol. 4 (2000), [12] A.I.Mal cev, On a class of homogeneous spaces, Izv. Akad. Nauk SSSR ser. Mat. 13 (1949), 9-32 [13] Ruh, Ernst A. Almost flat manifolds. J. Differential Geom. 17 (1982), no. 1, [14] Stong, R. E. On fibering of cobordism classes. Trans. Amer. Math. Soc. 178 (1973), [15] Shashidhar Upadhyay. A bounding question for almost flat manifolds. Trans. Amer. Math. Soc. 353 (2001), [16] S.T.Yau, Open problems in differential geometry, Proc. Sympo. in pure Math. 54 (1993), Department of Mathematics, Indiana University, Bloomington, IN47405, USA address: jfdavis@indiana.edu Nankai Institute of Mathematics, Weijin Road 94, Tianjin , P.R.China address: ffang@sun.nankai.edu.cn