The ER(n)-cohomology of BO(q), and real Johnson-Wilson orientations for vector bundles

TL;DR

This paper computes the ER(n)-cohomology ring of classifying spaces BO(q) using a spectral sequence and demonstrates the ER(n)-orientability of a specific Thom spectrum related to vector bundles.

Contribution

It explicitly calculates ER(n)^*(BO(q)) and establishes ER(n)-orientability of MO[2^{n+1}] spectrum, advancing understanding of real Johnson-Wilson theories.

Findings

Explicit computation of ER(n)^*(BO(q))

Demonstration of ER(n)-orientability of MO[2^{n+1}]

Identification of non-orientability at higher levels

Abstract

Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n)^*(BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2^{n+1}] is ER(n)-orientable (but not ER(n+1)-orientable), where MO[2^{n+1}] is defined as the Thom spectrum for the self map of BO given by multiplication by 2^{n+1}.