The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$

TL;DR

This paper computes the $ER(2)$-cohomology for specific classifying spaces and projective spaces, advancing the understanding of $ER(2)$-cohomology and its relation to TMF theories.

Contribution

It provides explicit calculations of $ER(2)$-cohomology for $B ext{Z}/(2^q)$ and $ ext{CP}^n$, extending to all Eilenberg-MacLane spaces and connecting to TMF_0(3).

Findings

Computed $ER(2)$-cohomology of $B ext{Z}/(2^q)$ and $ ext{CP}^n$.

Analyzed the Atiyah-Hirzebruch spectral sequence for $ER(2)^*( ext{CP}^ )$.

Connected $ER(2)$-cohomology to TMF_0(3) after completion.

Abstract

The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$ are computed along with the Atiyah-Hirzebruch spectral sequence for $ER(2)^*(\mathbb{C}P^\infty)$. This, along with other papers in this series, gives us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces. Since $ER(2)$ is $TMF_0(3)$ after a suitable completion, these computations also take care of that theory.