A characterization of the $\hat{A}$-genus as a linear combination of Pontrjagin numbers

TL;DR

This paper proves that any rational linear combination of Pontrjagin numbers that vanishes on certain spin manifolds with specific Ricci curvature conditions must be proportional to the $\hat{A}$-genus, refining previous results by Gromov and Lawson.

Contribution

It establishes a new characterization of the $\hat{A}$-genus as the unique linear combination of Pontrjagin numbers with specific vanishing properties on manifolds with nonnegative Ricci curvature.

Findings

Linear combinations vanishing on certain manifolds are multiples of the $\hat{A}$-genus.

Improves upon Gromov and Lawson's earlier results.

Combines ideas from Atiyah-Hirzebruch and Calabi-Yau theorem.

Abstract

We show in this short note that if a rational linear combination of Pontrjagin numbers vanishes on all simply-connected $4k$-dimensional closed connected and oriented spin manifolds admitting a Riemannian metric whose Ricci curvature is nonnegative and nonzero at any point, then this linear combination must be a multiple of the $\hat{A}$-genus, which improves on a result of Gromov and Lawson. Our proof combines an idea of Atiyah and Hirzebruch and the celebrated Calabi-Yau theorem.