Almost non-negative curvature and rational ellipticity in cohomogeneity two

Grove K.; Wilking B.; Yeager J.; Halperin S.

Abstract

An extension of a fundamental conjecture by R. Bott suggests that all simply connected closed almost non-negatively curved manifolds M are rationally elliptic, i.e., all but finitely many homotopy groups of such M are finite. We confirm this conjecture when in addition M supports an isometric action with orbits of codimension at most two. Our proof uses the geometry of the orbit space to control the topology of the homotopy fiber of the inclusion map of an orbit in M, and is applicable to more general contexts.