On characteristic numbers of $24$ dimensional String manifolds

TL;DR

This paper investigates the Pontryagin numbers of 24-dimensional String manifolds, providing explicit representatives of the String cobordism group and analyzing divisibility properties of characteristic numbers, with implications for topology and geometry.

Contribution

It constructs representatives of the 24-dimensional String cobordism group and determines divisibility properties of characteristic numbers, advancing understanding of String manifolds in topology.

Findings

Established 2-primary divisibility of the signature and modified signature.

Proved 3-primary divisibility of the twisted signature.

Provided explicit representatives of the String cobordism group at dimension 24.

Abstract

In this paper, we study the Pontryagin numbers of $24$ dimensional String manifolds. In particular, we find representatives of an integral basis of the String cobrodism group at dimension $24$, based on the work of Mahowald-Hopkins \cite{MH02}, Borel-Hirzebruch \cite{BH58} and Wall \cite{Wall62}. This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the $2$-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins-Singer \cite{HS05}, and also the $3$-primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner.