2002/09/13 by Burt Totaro, Totaro, Burt
Mathematics · #53C20 (Primary) 55P62 (Secondary) #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #math.AT #math.DG #msc:53C20 #msc:55P62
paper · pdf · doi:10.48550/arxiv.math/0209173
12 pages
arxiv created 2002/09/13 · arxiv updated 2009/11/30
Gromov showed that there is an upper bound on the Betti numbers of all closed Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether such manifolds (if simply connected) fall into only finitely many rational homotopy types. We give a negative answer, in fact in dimension 6, which is the smallest possible. We also give counterexamples to some related questions in dimensions 7 and 9, improving the original counterexamples by Fang and Rong which were in dimensions at least 22.