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Elliptic genus and string cobordism at dimension $24$

Fei Han, Ruizhi Huang

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.

Elliptic genus and string cobordism at dimension $24$

Abstract

It is known that spin cobordism can be determined by Stiefel-Whitney numbers and index theory invariants, namely -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.

Paper Structure

This paper contains 3 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['ellipticinjthm']}

Key Result

Theorem \oldthetheorem

The elliptic genus is injective and its image is a subgroup of $\mathbb{Z}[8\delta, \varepsilon]$ spanned by

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem: Theorem 1 and Corollary 3 in HH21
  • proof : Proof of Theorem \ref{['ellipticinjthm']}