category. Write $[ X,Y]_{G}$ for classes of based maps in the homotopy category of based maps.

A $G$-$CW$ complex $X$ is a topological union $X = \cup X_{n}$ of $G$-spaces such that $X_{0}$ is a disjoint union of orbits $G/H$ and $X_{n}$ is obtained from $X_{n-1}$ by attaching cells of the form $D^{n}\times G/H$ via attaching maps $\sigma:S^{n-1}\times G/H \to X_{n-1}$.

The equivariant Whitehead theorem holds for $G$-$CW$ complexes. That is, if $f:X\to Y$ is a $G$-equivariant weak equivalence between $G-CW$-complexes then $f$ is a $G$-homotopy equivalence.

A map $A\to X$ is said to have the *homotopy extension property* with respect to $Z$ if for any equivariant partial homotopy $H:X\times\{0\}\coprod_{A\times\{0\}}A\times I \to Z$ there is an equivariant map $H'$ making the diagram below commute $$\xymatrix{
X\times\{0\}\coprod_{A\times\{0\}}A\times I \ar[r]^-{H}\ar[d] & Z \\
X\times I \ar@{-->}[ur]^{\exists H'} & .
}$$ An equivariant *cofibration* $A\hookrightarrow X$ is an equivariant map which has the homotopy extension property with respect to all $Z$ in $\GTop$. Inclusions of sub-$G$-$CW$complexes $A\subseteq X$ are equivariant cofibrations.

Let $V$ be a real representation of $G$, write $S^{V}$ for the one-point compactification of $V$. The $V$th homotopy group of a based $G$-space $X$ is $$\pi_{V} X = [S^{V}, X]_{G} .$$ 