is an abelian group (with trivial $G$-action) then $H^{p,0}(X;\underline{A}) = H^{p}_{sing}(X/G; A)$. More generally, Borel and Bredon cohomology are related by the natural isomorphism $$H^{p,q}(X\times EG; \underline{A}) \iso H^{p+q}_{G}(X; A(q))$$ where $A(q)$ is $A$ with $\sigma$ acting by $(-1)^{q}$ (see [@DSLF:quat Proposition 1.15]).

Equivariant Dold-Thom theorem
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Let $X$ be a compactly generated Hausdorff space. The free abelian group on the points of $X$ is defined to be $ \mcal{Z}_{0}(X) = [\coprod_{d}SP^{d}(X)]^{+}$, where $SP^{d}(X)$ is the $d$th symmetric product on $X$ and $(-)^{+}$ denotes group completion of the displayed monoid which is topologized via the quotient topology.

The degree homomorphism $\deg:\mcal{Z}_{0}(X) \to \Z$ is defined by $\deg(\sum n_{i}x_i) = \sum n_i$ is a continuous homomorphism. Write $\mcal{Z}_{0}(X)_{0}$ for the kernel of this map. Notice that there is an isomorphism of topological groups $\mcal{Z}_{0}(X) \iso \mcal{Z}_{0}(X_+)_{0}$.

If $X$ is a $G$-space then the action on $X$ induces a $G$-action on $\mcal{Z}_{0}(X)$ and $\mcal{Z}_{0}(X)_{0}$. By [@LF:Gequiv Corollary 2.9] when $X$ is a $G$-$CW$ complex so is $\mcal{Z}_{0}(X)$.

The classical Dold-Thom theorem says that $\pi_{n}\mcal{Z}_{0}(X)_{0} = \tilde{H}_{n}(X;\Z)$ and the equivariant Dold-Thom theorem proved by Lima-Filho [@LF:Gequiv] and dos Santos [@DS:equiDT] says that $$\pi_{V}\mcal{Z}_{0}(X)_{0} = \tilde{H}_{V}(X;\underline{\Z}).$$ In particular $\mcal{Z}_{0}(S^{V})_{0}$ is an Eilenberg-Maclane space $K(\underline{Z}, 