Topological properties of positively curved manifolds with symmetry

TL;DR

This paper establishes upper bounds on the Euler characteristic of positively curved manifolds with large symmetry groups, providing new obstructions to certain symmetric spaces and manifold constructions admitting positive curvature.

Contribution

It introduces bounds on Euler characteristics for positively curved manifolds with large isometric torus actions, advancing understanding of their topological constraints.

Findings

Upper bounds for Euler characteristic with large symmetry

Obstructions to symmetric spaces with positive curvature

Restrictions on manifold products and connected sums

Abstract

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.