Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

TL;DR

This paper proves that closed manifolds with almost nonnegative curvature operator and bounded curvature have nonnegative Euler characteristic, extending known results from Ricci curvature to the curvature-operator setting.

Contribution

It establishes nonnegativity of Euler characteristic under almost nonnegative curvature operator and bounded curvature, and extends vanishing results to this setting.

Findings

Euler characteristic is nonnegative under given conditions.

Vanishing of Euler characteristic, signature, and $\widehat{A}$-genus for infinite fundamental group.

Extension of results from Ricci curvature to curvature-operator context.

Abstract

We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann--Sebastian--Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed $2n$-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the $\widehat{A}$-genus, extending recent work of Chen--Ge--Han from almost nonnegative Ricci curvature to the curvature-operator setting.