Z}}$-basis of $C_{{n}}X$ is given by the set of nondegenerate ${{n}}$-dimensional simplices of $X$. If $X$ is equipped with a free simplicial action of $G$, then this induces an action of $G$ on $C_*X$ by chain maps, and a ${{{\mathbb Z}G}}$-basis for each $C_{{n}}X$ is given by a collection of nondegenerate distinguished ${{n}}$-dimensional simplices of $X$, one from each $G$-orbit.

If $X$ is locally effective as defined above, then so is $C_*X$ (for evaluating the differential, we observe that a simplex $x$ is degenerate if and only if $x=s_{{i}}d_{{i}}x$ for some ${{i}}$, and this can be checked algorithmically).

\[con:G\_action\_loc\_eff\] We fix a finite group $G$. All simplicial sets are locally effective, equipped with a free action of $G$ and cellular in the locally effective sense. All chain complexes are non-negatively graded locally effective chain complexes of free ${{{\mathbb Z}G}}$-modules that are moreover cellular in the locally effective sense.

All simplicial maps, chain maps, chain homotopies, etc. are equivariant and computable.

Later, Convention \[con:fibrewise\] will introduce additional standing assumptions.

An *effective* chain complex is a (locally effective) chain complex equipped with an algorithm that generates a list of elements of the distinguished basis in any given dimension (in particular, the distinguished 