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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 coinduce
We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equi
We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This lea
For a finite group $G$, there is a map $RO(G) to {rm Pic}(Sp^G)$ from the real representation ring of $G$ to the Picard group of $G$-spectra. This map is not known to be surjective in general, but we prove that when $G$ is cyclic this map is indeed s
Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining the model
We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.