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On the topology of manifolds with positive isotropic curvature. (English) Zbl 1166.53026
The authors show that a closed orientable Riemannian $$n$$-manifold, $$n\geq 5$$, with positive isotropic curvature and free fundamental group is homeomorphic to a connected sum of copies of $$S^{n-1}\times S^1$$.
Let $$(M,g)$$ be a closed, orientable, Riemannian manifold with positive isotropic curvature. By a paper of M. J. Micallef and J. D. Moore [Ann. Math. (2) 127, No. 1, 199–227 (1988; Zbl 0661.53027)], if $$M$$ is simply connected, then $$M$$ is homeomorphic to a sphere of the same dimension. The authors generalize this to the case when the fundamental group $$M$$ is a free group.
Theorem 1.1. Let $$M$$ be a closed, orientable Riemannian $$n$$-manifold with positive isotropic curvature. Suppose that $$\pi_1(M)$$ is a free group on $$k$$ generators. Then, if $$n\neq 4$$ or $$k=1$$ (i.e., $$\pi_1(M)=\mathbb{Z}$$), $$M$$ is homeomorphic to the connected sum of $$k$$ copies of $$S^{n-1}\times S^1$$.
The proof of this result is based on a theorem of M. Micaleff and J. Moore (op. cit.) which states that for a closed manifold $$M$$ with positive isotropic curvature, $$\pi_i(M)=0$$ for $$2\leq i\leq\frac{n}{2}$$ and the following purely topological result established in the present paper:
Theorem 1.3. Let $$M$$ be a smooth, orientable, closed $$n$$-manifold such that $$\pi_1(M)$$ is a free group on $$k$$ generators and $$\pi_i(M)=0$$ for $$2\leq i\leq\frac{n}{2}$$. If $$n\neq 4$$ or $$k=1$$, then $$M$$ is homeomorphic to the connected sum of $$k$$ copies of $$S^{n-1}\times S^1$$.
Reviewer: Ioan Pop (Iaşi)

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 57M05 Fundamental group, presentations, free differential calculus
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##### References:
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