for pairs $(C_*,D_*)$ of chain complexes is introduced similarly using strong equivalences of pairs. A simplicial set $X$ is equipped with *effective homology* if $C_*X$ is. Finally, a pair $(X,A)$ of simplicial sets is equipped with *effective homology* if $(C_*X,C_*A)$ is.[^11]

In what follows, we will only assume $(X,A)$, $Y$, $B$ to be equipped with effective homology. Consequently, it can be seen that Theorems \[theorem:equivariant\] and \[thm:main\_theorem\] also hold under these weaker assumptions. The dimension restriction on $X$ can be weakened to: the equivariant cohomology groups of $(X,A)$, defined in Section \[sec:equiv\_cohomology\], vanish above dimension $2{{d}}$.[^12]

The following theorem shows that, in order to equip a chain complex with effective homology, it suffices to have it equipped with effective homology in the non-equivariant sense.

\[t:vokrinek\] Let $C_*$ be a chain complex (of free ${{{\mathbb Z}G}}$-modules). Suppose that as a chain complex of abelian groups $C_*$ can be equipped with effective homology (i.e. in the non-equivariant sense). Then it is possible to equip $C_*$ with effective homology in the equivariant sense. This procedure is algorithmic.

The original strong equivalence $C_*{\xLeftrightarrow{\ \ \ }}C_*^{\mathrm{ef}}$ gets replaced by an equivariant one $C_*{\xLeftrightarrow{\ \ \ }}BC_*^{\mathrm{ef}}$, where $BC_*^{\mathrm{ef}}$ is a bar construction of some 