Positive curvature and rational ellipticity

Abstract

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm

Bounds on Sectional Curvature and Interactions with Topology

“…The conjecture was confirmed for the known simply-connected even-dimensional examples of positive curvature in [4] and for the known simply-connected odddimensional examples in [31]. It seems that this conjecture is very geometrical in nature lacking the connection to topology on which we focus in this survey.…”

“…. #CP 2 for k ≥ 3 has second Betti number k. The corresponding Betti number of the 4-dimensional torus is 4 2 = 6. Hence Gromov's conjecture is infringed for k ≥ 7.…”

“…First one specifies a biquotient action of U(1) 4 in such a way that the resulting 8-dimensional quotient M satisfies…”

Homotopy Invariants and Almost Non-Negative Curvature

“…If these conjectures are true, then it is possible that in an ANSC-bundle F Ñ M Ñ N , the fibre could be an F 0 -space (see Subsection 3.1). (In [1], the authors study such manifolds as total spaces of fibrations. )…”

Soft Restrictions on Positively Curved Riemannian Submersions

“…53C20. Both authors were supported by research grants MTM2011-22612 from the Ministerio de Ciencia e Innovación (MCINN) and MINECO: ICMAT Severo Ochoa project SEV-2011-0087; the first author was also suppported by FPI grant BES-2012-053704 1.…”