Note that $S^{V}$ always has at least two fixed points, $0$ and $\infty$.

When $G=\Z/2$ and $V = \R^{p,q}$, where $V =\R^{p+q}$ with $G$ acting trivially on the first $p$-components and on the last $q$-components the $G$ action is multiplication by $-1$ we use the notation $$\pi_{p,q}X = \pi_{\R^{p,q}}X .$$

Borel homology and cohomology
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The *Borel-equivariant cohomology* of $X$ with coefficients in an abelian group $A$ is defined to be the ordinary singular cohomology of the homotopy orbit space of $X$: $$H^{p}_{G}(X; A) = H^{p}((X\times EG)/G ; A).$$ Similarly the *Borel-equivariant homology* is defined to be $$H_{p}^{G}(X; A) = H_{p}((X\times EG)/G ; A).$$

When $X$ has free $G$-action then $ (X\times EG)/G \to X/G$ is a homotopy equivalence and therefore when $X$ has free $G$-action $ H^{p}(X/G; A) \iso H^{p}_{G}(X; A)$ and $H_{p}(X/G; A) \iso H_{p}^{G}(X; A)$.

Mackey functors
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Bredon homology and cohomology take Mackey functors as coefficients. There are several equivalent ways to define a Mackey functor [@May:equi]. Classically for $G$ a finite group one defines a Mackey functor as follows . Let $\mcal{F}_{G}$ denote the category of finite $G$-sets as objects and with equivariant set maps as morphisms. A *Mackey functor* $\underline{M}$ consists of a pair 