Landweber flat real pairs and ER(n)-cohomology

TL;DR

This paper leverages the algebraic structure of the Bockstein spectral sequence to compute ER(n)-cohomology of certain spaces, extending known results to a broader class of spaces and providing explicit calculations.

Contribution

It demonstrates that ER(n)^*(Z) can be derived from E(n)^*(Z) via base change for spaces in Landweber flat real pairs, enabling new cohomology computations.

Findings

Computed ER(n)^*(Z) for Eilenberg-MacLane spaces K(Z, 2m+1), K(Z/2^q, 2m), K(Z/2, m)

Extended cohomology calculations to connective covers of BO, BSO, BSpin, and BO<8>

Established a method to derive ER(n)^*(Z) from E(n)^*(Z) using spectral sequence structure

Abstract

We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to ER(n)^*(pt) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E(n), the cohomology ring ER(n)^*(Z) can be obtained from E(n)^*(Z) by base change. In particular, our results allow us to compute the Real Johnson-Wilson cohomology of the Eilenberg-MacLane spaces Z = K(Z, 2m+1), K(Z/2^q, 2m), K(Z/2, m) for all natural numbers $m$ and $q$, as well as connective covers of BO: BO, BSO, BSpin, and BO<8> (the last for n<3 only).