Curvature operators and rational cobordism

TL;DR

This paper establishes linear eigenvalue inequalities for curvature operators that ensure the vanishing of certain rational cobordism invariants on closed Riemannian spin manifolds, linking curvature conditions to topological invariants.

Contribution

It introduces new curvature inequalities that guarantee the vanishing of various rational cobordism invariants, extending the understanding of geometric-topological relations.

Findings

Derived inequalities imply vanishing of the twisted  genus.

Established surgery-stable curvature conditions for multiple invariants.

Connected curvature conditions to the rational cobordism class.

Abstract

We determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted $\hat A$ genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors. These inequalities yield surgery-stable curvature conditions tailored to annihilate further rational cobordism invariants, such as the Witten genus, elliptic genus, signature, and even the rational cobordism class itself.