On the hyperreal dual Steenrod algebra

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 $H\underline{\mathbb F}_2\otimes H\underline{\mathbb Z}$.

💡 Research Summary

The paper “On the hyperreal dual Steenrod algebra” by Michael A. Hill and Michael J. Hopkins develops a comprehensive framework for computing the dual Steenrod algebra in the equivariant setting of the cyclic group $G=C_{2^{n}}$, using Bredon homology with constant Mackey functor coefficients $\underline{\mathbb Z}$ and $\underline{\mathbb Z}/2$. The central object of study is the $G$‑equivariant spectrum $\Xi_n = MU^{((G))}=N_{C_{2^{n}}}^{C_{2}}MU_{\mathbb R}$, the norm of the Real cobordism spectrum. The authors exploit the Reduction Theorem (originally proved in