category $\GTop$ of $G$-spaces consists of compactly generated spaces equipped with a left $G$-action and morphisms are continuous $G$-equivariant maps. If $X$ is a $G$-space and $H\subseteq G$ is a subgroup write $X^{H}$ for the subspace of all points fixed by $H$. The category $\GTop_*$ of based $G$-spaces consists of $G$-spaces $X$ together with a $G$-invariant basepoint $x\in X$ and maps are base-point preserving equivariant maps. A space together with a disjoint, invariant base-point will be denoted $X_{+}$.

Equivariant homotopy theory
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Let $I$ denote the unit interval with trivial $G$-action. A $G$-homotopy between two equivariant maps $f,g:X \to Y$ is an equivariant map $F:X\times I \to Y$ such that $F|_{X\times\{0\}} = f$ and $F|_{X\times\{1\}} = g$. An equivariant map $f:X\to Y$ is an *equivariant homotopy equivalence* provided there is an equivariant map $g:Y \to X$ such that both $f\circ g $ and $g \circ f$ are $G$-equivalently homotopic to the identity. An equivariant map $f:X\to Y$ is a *$G$-weak equivalence* provided both $f^{H}:X^{H}\to Y^{H}$ is a non-equivariant weak equivalence for all subgroups $H\subseteq G$. Formally inverting the $G$-weak equivalences gives the homotopy category of $G$-spaces. Similarly inverting the based $G$-weak equivalences between based $G$-spaces we obtain the based $G$-homotopy 