Elliptic genus and string cobordism at dimension $24$

Fei Han and Ruizhi Huang

Vol. 328 (2024), No. 2, 275–286

DOI: 10.2140/pjm.2024.328.275

Abstract

It is known that spin cobordism can be determined by Stiefel–Whitney numbers and index theoretic invariants, namely $\mathrm{KO}<!--FUNCTION\; APPLICATION-->$-theoretic Pontryagin numbers. We show that string cobordism at dimension 24 can be determined by elliptic genus, a higher index theoretic 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.

Keywords

string cobordism, elliptic genus, string manifold, nonpositive Ricci curvature, Witten genus

Mathematical Subject Classification

Primary: 57R20, 58J26

Secondary: 53C21, 57R90

Milestones

Received: 28 February 2023

Revised: 20 October 2023

Accepted: 30 March 2024

Published: 30 April 2024