OBSTRUCTIONS TO FREE ACTIONS ON BAZAIKIN SPACES
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Springer Science+Business Media New York (2020)
OBSTRUCTIONS TO FREE ACTIONS ON BAZAIKIN SPACES E. KHALILI SAMANI Department of Mathematics Syracuse University Syracuse, NY 13244, USA [email protected]
Abstract. Apart from spheres and an infinite family of manifolds in dimension seven, Bazaikin spaces are the only known examples of simply connected Riemannian manifolds with positive sectional curvature in odd dimensions. We consider positively curved Riemannian manifolds whose universal covers have the same cohomology as Bazaikin spaces and prove structural results for the fundamental group in the presence of torus symmetry.
1. Introduction An important question in Riemannian geometry is to investigate the structure of fundamental groups of Riemannian manifolds with non-negative sectional curvature. A well-known example of this is a theorem of Gromov which states that the fundamental group of a complete Riemannian manifold M n with non-negative sectional curvature has at most C(n) generators, where C(n) is a constant depending only on the dimension of M (see [Gro78]). In addition, the Cheeger–Gromoll splitting theorem, together with a theorem of Wilking, implies that a group G is the fundamental group of a non-negatively curved Riemannian manifold if and only if G has a normal subgroup isomorphic to Zd such that the quotient group is finite (see [CG71] and [Wil00, Thm. 2.1]). Under the stronger assumption of positive curvature, the only known further obstructions are the results of Bonnet–Myers and Synge which together imply that the fundamental group of a positively curved Riemannian manifold is finite and, moreover, trivial or Z2 if the dimension of the manifold is even. As for examples, the largest class of groups which arise as fundamental groups of positively curved manifolds are the spherical space form groups. These are groups that act freely and linearly on spheres (for a complete classification, see [Wol11, Chap. III]). The first step in the classification of spherical space form groups is to establish that they satisfy the (p2 ) and (2p) conditions, which mean respectively that every subgroup of order p2 or 2p is cyclic. The (p2 ) condition was proved by Smith for groups acting freely on a mod p homology sphere, i.e., a space whose homology groups with coefficients in Zp coincide with that of a sphere (see [Smi44]). Moreover, the (2p) condition holds for groups acting freely on a mod 2 homology DOI: 10.1007/S00031-020-09625-w Received February 11, 2020. Accepted August 10, 2020. Corresponding Author: E. Khalili Samani, e-mail: [email protected]
E. KHALILI SAMANI
sphere by results of Milnor and Davis (see [Mil57] and [Dav83]). In 1965, Chern asked if the (p2 ) condition holds for the fundamental groups of Riemannian manifolds with positive sectional curvature. This question was not answered for over 30 years until Shankar proved that there are examples for which the (p2 ) condition fails for p = 2 and the (2p) condition fails for all p (see [Sha98]). Later, Bazaikin and Grove–Shankar (see [Baz99] an
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