  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 $BBU_{\mathbb R}$.
