Nonnegative curvature, elliptic genus and unbounded Pontryagin numbers

TL;DR

This paper investigates the cobordism classification of spin manifolds with nonnegative sectional curvature, revealing infinite cobordism types in certain dimensions and exploring potential obstructions to nonnegative curvature.

Contribution

It demonstrates the existence of infinitely many cobordism types of simply connected, nonnegatively curved spin manifolds in dimensions 12 and higher, and discusses obstructions to nonnegative curvature.

Findings

Infinite cobordism types in dimensions 12 and above

Existence of obstructions to nonnegative curvature

Analysis of elliptic genus and Pontryagin numbers

Abstract

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to nonnegative curvature.