zbMATH — the first resource for mathematics

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)

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
57M05 Fundamental group, presentations, free differential calculus
Full Text: DOI arXiv
[1] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.
[2] Ailana M. Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. (2) 158 (2003), no. 1, 345 – 354. · Zbl 1044.53023 · doi:10.4007/annals.2003.158.345 · doi.org
[3] Ailana Fraser and Jon Wolfson, The fundamental group of manifolds of positive isotropic curvature and surface groups, Duke Math. J. 133 (2006), no. 2, 325 – 334. · Zbl 1110.53027 · doi:10.1215/S0012-7094-06-13325-2 · doi.org
[4] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993) Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1 – 213. · Zbl 1262.90126 · doi:10.1007/s10107-010-0354-x · doi.org
[5] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001
[6] Heinz Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257 – 309 (German). · Zbl 0027.09503 · doi:10.1007/BF02565622 · doi.org
[7] M. Kreck and W. Lück, Topological rigidity for non-aspherical manifolds, to appear in Quarterly Journal of Pure and Applied Mathematics. · Zbl 1196.57018
[8] Mario J. Micallef and McKenzie Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), no. 3, 649 – 672. · Zbl 0804.53058 · doi:10.1215/S0012-7094-93-07224-9 · doi.org
[9] Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199 – 227. · Zbl 0661.53027 · doi:10.2307/1971420 · doi.org
[10] J. H. C. Whitehead, On simply connected, 4-dimensional polyhedra, Comment. Math. Helv. 22 (1949), 48 – 92. · Zbl 0036.12704 · doi:10.1007/BF02568048 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.