The Cayley plane and String bordism

TL;DR

This paper characterizes the kernel of the Witten genus in string bordism as generated by Cayley plane bundles, linking Lie group calculations with BP-Hopf-ring properties at primes greater than 3.

Contribution

It establishes a precise algebraic description of the Witten genus kernel in string bordism using Cayley plane bundles and divisibility properties.

Findings

Kernel of Witten genus equals Cayley plane bundle classes (away from 6)

Divisibility properties match between Lie group calculations and BP-Hopf-ring calculations

Results hold for primes greater than 3

Abstract

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle characteristic numbers arising in Borel-Hirzebruch Lie-group-theoretic calculations correspond precisely to the divisibility properties arising in the Hovey-Ravenel-Wilson BP-Hopf-ring-theoretic calculation of string bordism at primes >3.