Freeness and equivariant stable homotopy

We introduce a notion of freeness for (Formula presented.) -graded equivariant generalized homology theories, considering spaces or spectra (Formula presented.) such that the (Formula presented.) -homology of (Formula presented.) splits as a wedge of the (Formula presented.) -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 (Formula presented.) -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 (Formula presented.) and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various (Formula presented.) pushouts, and we apply this to describe the homology of (Formula presented.).