On the hyperreal dual Steenrod algebra

TL;DR

This paper computes the dual Steenrod algebra in an equivariant setting for certain Bredon homology theories and uses spectral sequences to analyze the RO-graded homotopy of specific Eilenberg–Mac Lane spectra.

Contribution

It provides the first computation of the dual Steenrod algebra for Bredon homology with constant coefficients in an equivariant context and develops spectral sequences for RO-graded homotopy.

Findings

Computed the dual Steenrod algebra for Bredon homology with constant coefficients.

Developed an equivariant Greenlees–Serre spectral sequence.

Provided a spectral sequence for RO-graded homotopy of specific spectra.

Abstract

We compute the dual Steenrod algebra for Bredon homology with constant coefficients $\underline{\mathbb Z}$ and $\underline{\mathbb Z}/2$ in the category of modules over $MU^{((G))}$, the norm to $G=C_{2^n}$ of $MU_{\mathbb R}$. Using this and an equivariant version of the Greenlees--Serre spectral sequence, we give a spectral sequence computing the $RO$-graded homotopy of the Eilenberg--Mac Lane spectrum $H\underline{\mathbb F}_2\otimes H\underline{\mathbb Z}$.