On the topology of manifolds with positive isotropic curvature

We show that a closed orientable Riemannian $n$-manifold, $n \ge 5$, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of $S^{n-1} \times S^1$.


Introduction
Let (M, g) be a closed, orientable, Riemannian manifold with positive isotropic curvature. By [9], if M is simply-connected then M is homeomorphic to a sphere of the same dimension. We shall generalise this to the case when the fundamental group of M is a free group. Theorem 1.1. Let M be a closed, orientable Riemannian n-manifold with positive isotropic curvature. Suppose that π 1 (M ) is a free group on k generators. Then, if n = 4 or k = 1 (i.e. π 1 (M ) = Z), M is homeomorphic to the connected sum of k copies of S n−1 × S 1 .
We note that a conjecture of M. Gromov( [4] Section 3 (b)) and A. Fraser [2], based on the work of Micallef-Wang [8], states that any compact manifold with positive isotropic curvature has a finite cover satisfying our hypothesis.
Conjecture 1 (M. Gromov-A. Fraser). π 1 (M ) is virtually free, i.e., it is a finite extension of a free group.
It is known by the work of A. Fraser [2] and A. Fraser and J. Wolfson [3] that π 1 (M ) does not contain any subgroup isomorphic to the fundamental group of a closed surface of genus at least one.
Our starting point is the following fundamental result of M. Micallef and J. Moore [9].
It is clear that the following purely topological result, together with the Micallef-Moore theorem, implies Theorem 1.1 Theorem 1.3. Let M be a smooth, orientable, closed n-manifold such that π 1 (M ) is a free group on k generators and π i (M ) = 0 for 2 ≤ i ≤ n 2 . If n = 4 or k = 1, then M is homeomorphic to the connected sum of k copies of S n−1 × S 1 . Henceforth let M be a smooth, orientable, closed n-manifold such that π 1 (M ) is a free group on k generators and π i (M ) = 0 for 2 ≤ i ≤ n 2 . We assume throughout that all manifolds we consider are orientable.
Let M be the universal cover of M . Hence π 1 ( M ) is trivial and so is π i ( M ) = π i (M ) for 2 ≤ i ≤ n 2 . We shall show that the homology of M is isomorphic as π 1 (M )-modules to that of the connected sum of k copies of S n−1 × S 1 . We then show that M is homotopy equivalent to the connected sum of k copies of S n−1 × S 1 using Theorems of Whitehead. Finally, recent results of Kreck and Lück allow us to conclude the result.
Acknowledgements. We thank the referees for helpful comments and for suggesting a simplification of our proof.

The homology of M
Let X denote the wedge ∨ k j=1 S 1 of k circles and let x denote the common point on the circles. Choose and fix an isomorphism ϕ from π 1 (M, p) to π 1 (X, x) for some basepoint p ∈ M . We shall use this identification throughout. Denote π 1 (M, p) = π 1 (X, x) by π.
As X is an Eilenberg-Maclane space, there is a map f : (M, p) → (X, z) inducing ϕ on fundamental groups and a map s : We deduce the homology of M using the Hurewicz Theorem and Poincaré duality.
We deduce the homology in dimensions above n/2 using Poincaré duality for M with coefficients in the module Z[π], namely . Hence Poincaré duality with coefficients in Z[π] is the same as Poincaré duality for a non-compact manifold relating homology to cohomology with compact support.
To apply Poincaré duality, we need the following lemma.
Proof. As the map s induces an isomorphism on homotopy groups in dimensions at most n/2, it induces isomorphisms on the cohomology groups with twisted coefficients. Specifically, we can add cells of dimensions k ≥ n/2 + 2 to M to obtain an Eilenberg-MacLane spaceM for the group π, which is thus homotopy equivalent to X. For i ≤ n/2 and any Z[π]-module A, it follows that where the first equality follows as the cells added to M to obtainM are of dimension at least n/2 + 2 and the second as the spaces are homotopy equivalent.
By applying Poincaré duality, we obtain the following result. Lemma 2.3. Let M be a smooth, orientable, closed n-manifold such that π 1 (M ) is a free group on k generators and π i (M ) = 0 for 2 ≤ i ≤ n 2 . Then, for the universal cover M of M , , where X is the universal cover of X, determined by the isomorphisms s * : π 1 (X, z) → π 1 (M, p) on fundamental groups.

Homotopy type
We now show that M is homotopy equivalent to the connected sum Y of k copies of S n−1 × S 1 . Our first step is to construct a map g : Y → M . We shall then show that it is a homotopy equivalence.
Note that Y has the structure of a CW-complex obtained as follows. The 1skeleton of Y is the wedge X of k circles. Let α i denote the ith circle with a fixed orientation.
We attach k (n − 1)-cells D j , with the jth attaching map mapping ∂D n−1 to the midpoint x j of the jth circle. Finally, we attach a single n-cell ∆.
We associate to D j an element A j ∈ π n−1 (Y, x). Namely, as the attaching map is constant, the jth (n − 1)-cell gives an element B j ∈ π n−1 (Y, x j ). We consider the subarc β j of α j joining z j to x in the negative direction and let A j be obtained from B j by the change of basepoint isomorphism using β j .
Note that if we instead chose the arc joining z j to x in the positive direction, then the resulting element is −α j · A j . By the construction of Y , it follows that the attaching map of the (n − 1)-cell represents the element in π n−1 (Y ) regarded as a module over π 1 (Y ). This can be seen for instance by using Poincaré duality.
We now construct the map g : Y → M . Recall that we have a map s : (X, z) → (M, p) inducing the isomorphism ϕ −1 on fundamental groups. We define g on the 1-skeleton X of Y by g| X = s. We henceforth identify the fundamental groups of Y and M using the isomorphism ϕ, i.e., π 1 (Y, z) is identified with π.
We next extend g to the n-cell of Y as follows. By Hurewicz theorem and Lemma 2.3, we have isomorphisms of π-modules π n−1 (M, p) = H n−1 ( M , Z) and π n−1 (Y, z) = H n−1 ( M , Z). By Lemma 2.3, each of these modules is isomorphic to H 1 c ( X, Z) with the isomorphisms determined by the identifications of the fundamental groups.
Under the above isomorphisms the elements A j correspond to elements A ′ j in π n−1 (M, p). Consider the element B ′ j of π n−1 (M, g(z j )) obtained from A ′ j by the basechange map using the arc f (β j ). We define the map g on D j extending the constant map on its boundary to be a representative of B j .
As the π-modules π n−1 (M, p) and π n−1 (Y, z) are isomorphic, the image g of ∂∆ is homotopically trivial. Hence we can extend the map g across the cell ∆. Proof. Let G : Y → M be the induced map on the universal covers. By Lemma 2.3 applied to M and Y , we see that H p ( Y ) = H p ( M ) = 0 for 0 < p = n − 1 and G induces an isomorphism on H n−1 . Thus the map G is a homology equivalence. By a theorem of Whitehead [10], a homology equivalence between simply-connected CW-complexes is a homotopy equivalence.
It follows that G induces isomorphisms G * : π k ( Y ) → π k ( M ) for k > 1. As covering maps induce isomorphims on higher homotopy groups, and g induces an isomorphism on π 1 , it follows that g is a weak homotopy equivalence, hence a homotopy equivalence(see [5]).

Proof of Theorem 1.3
The rest of the proof of Theorem 1.3 is based on results of Kreck-Lück [7]. In [7], the authors define a manifold N to be a Borel manifold if any manifold homotopy equivalent to N is homeomorphic to N . We have shown that a manifold M satisfying the hypothesis of Theorem 1.3 is homotopy equivalent to the connected sum Y of k copies of S n−1 × S 1 . Hence it suffices to observe that Y is Borel.
By Theorem 0.13(b) of [7], the manifold S n−1 × S 1 is Borel for n ≥ 4. This completes the proof in the case when π 1 (M ) = Z. Further, if n ≥ 5, then Theorem 0.9 of [7] says that the connected sum of Borel manifolds is Borel, hence Y is Borel. This concludes the proof for π 1 (M ) a free group and n ≥ 5.
Finally, in the case when n = 3 by the Knesser conjecture (proved by Stallings) the manifold M is a connected sum of manifolds whose fundamental group is Z. As M is orientable, it follows that if M is expressed as a connected sum of prime manifolds (such a decomposition exists and is unique by the Knesser-Milnor theorem), then each prime component is either S 2 ×S 1 or a homotopy sphere. By the Poincaré conjecture (Perelman's theorem), every homotopy 3-sphere is homeomorphic to a sphere. It follows that M is the connected sum of k copies of S 2 × S 1 .