Freeness and equivariant stable homotopy

TL;DR

This paper introduces a new notion of freeness for $RO$-graded equivariant homology theories, simplifying computations and providing explicit examples, especially for spectra like $BU_{ ext{R}}$, with applications to $E_{ ext{infty}}$ pushouts.

Contribution

It defines a class of free spectra in equivariant homology, showing they are closed under key operations and facilitate easier calculations, with detailed examples and new computational tools.

Findings

Freeness simplifies $RO$-graded equivariant homology computations.

The class of free spectra is closed under basic equivariant operations.

Explicit calculations for $BU_{ ext{R}}$ and related spectra demonstrate the theory's effectiveness.

Abstract

We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, the $RO$-graded Bredon homology is extremely easy to compute, and if these spaces have additional structure, then this can also be easily determined. In particular, the homology of a space with this property naturally has the structure of a co-Tambara functor (and compatibly with any additional product structure). We work this out in the example of $BU_{\mathbb R}$ and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various $E_{\infty}$ pushouts, and we apply this to describe the homology of $BBU_{\mathbb R}$.