Elliptic genus and string cobordism at dimension $24$

TL;DR

This paper demonstrates that elliptic genus uniquely determines 24-dimensional string cobordism and computes its image, leading to geometric implications under curvature conditions.

Contribution

It establishes that elliptic genus characterizes 24-dimensional string cobordism and computes its image, extending the understanding of cobordism invariants beyond classical invariants.

Findings

Elliptic genus determines 24-dimensional string cobordism.

Computed the image of 24-dimensional string cobordism under elliptic genus.

Under certain curvature conditions, manifolds bound string manifolds.

Abstract

It is known that spin cobordism can be determined by Stiefel-Whitney numbers and index theory invariants, namely $KO$-theoretic Pontryagin numbers. In this paper, we show that string cobordism at dimension 24 can be determined by elliptic genus, a higher index theory invariant. We also compute the image of 24 dimensional string cobordism under elliptic genus. Using our results, we show that under certain curvature conditions, a compact 24 dimensional string manifold must bound a string manifold.