On the hyperreal dual Steenrod algebra
Michael A. Hill, Michael J. Hopkins
TL;DR
The paper develops a thorough equivariant framework to compute the $RO(G)$-graded dual Steenrod algebra associated to Bredon homology with coefficients $\underline{\mathbb Z}$ and $\underline{\mathbb Z}/2$ in the setting of $MU^{((G))}$-modules. It implements the Reduction Theorem to realize $H\underline{\mathbb Z}$ as a twisted monoid-ring module over a finite-$G$-indexed algebra $A$, enabling explicit smash-product and additive-structure descriptions, and then extends to a multiplicative theory via an equivariant Hahn–Wilson-type operadic analysis. The relative theory is organized around the relative dual Steenrod algebra, with the additive structure described by a Hopf algebroid on $(\pi_{\underline{\star}}H\underline{\mathbb Z},\pi_{\underline{\star}}(H\underline{\mathbb Z}\otimes_{\Xi_n}H\underline{\mathbb Z}))$, and the multiplicative structure controlled by norm-compatible extensions, induced/classified by subgroups and their fixed points. Finally, the authors formulate an equivariant Greenlees--Serre spectral sequence for the absolute case and demonstrate a concrete $C_2$ instance that recovers and augments Hu–Kriz results, illustrating the viability of the approach for RO-graded equivariant algebra computations. The work provides a coherent method to derive explicit additive and multiplicative structures of relative and absolute equivariant dual Steenrod algebras and offers a practical route to RO-graded computations in equivariant stable homotopy theory.
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 $H\underline{\mathbb F}_2\otimes H\underline{\mathbb Z}$.