The Cayley plane and string bordism

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 greater than $3$.