Freeness and equivariant stable homotopy

Abstract

We introduce a notion of freeness for RO$RO$‐graded equivariant generalized homology theories, considering spaces or spectra E$E$ such that the R$R$‐homology of E$E$ splits as a wedge of the R$R$‐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$RO$‐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 BUR$BU_{\mathbb {R}}$ and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various E∞$E_{\infty }$ pushouts, and we apply this to describe the homology of BBUR$BBU_{\mathbb {R}}$.

The RO(C4) cohomology of the infinite real projective space

“…It is the minimal decomposition that we have used as input for the computer program of [2]. 7 The spectral sequence for B C 4 † 2 Applying k F on the filtration of B C 4 † 2C gives a spectral sequence…”

“…Another application of the Hu-Kriz computation is the definition of equivariant Dyer-Lashof operations by Wilson [17] in the F 2 -homology of C 2 -spectra with symmetric multiplication. Many of these results rely on the homology of certain spaces being free as modules over the homology of a point, and there is a robust theory of such free spectra, described in Hill [7].…”