The real Brown-Peterson homology of $\Omega^\rho S^{\rho + 1}$

TL;DR

This paper computes the $RO(C_2)$-graded real Brown-Peterson homology of a specific equivariant loop space, extending classical results to a $C_2$-equivariant setting and developing new algebraic tools.

Contribution

It provides the first $C_2$-equivariant computation of Brown-Peterson homology for a representation-loop space and introduces comodule Nishida relations for $ ho$-loop spaces.

Findings

Computed the $RO(C_2)$-graded real Brown-Peterson homology of $ ho$-loop spaces.

Established comodule Nishida relations for $ ho$-loop spaces.

Extended classical homology computations to an equivariant context.

Abstract

We compute the $RO(C_2)$-graded real Brown--Peterson homology of the representation-loop space $\Omega^\rho S^{\rho + 1}$, where $\rho$ is the regular representation of the cyclic group of order two. This calculation gives a $C_2$-equivariant analogue of the classical computation of Brown--Peterson homology of the double loop space $\Omega^2 S^3$ due to Ravenel. Along the way, we develop comodule Nishida relations for $\rho$-loop spaces.