For a CW complex \(X\) with cells only in even dimensions, the integral homology \(H_n(X,\mathbb{Z})\) is a free abelian group for any integer \(n\geq 0\). The main goal of the memoir under review is to prove a \(G\)-equivariant version of this result. For a compact Lie group G, the authors consider \(\text{Rep}^*(G)\)-cell complexes: \(G\)-spaces \(X\) with increasing sequences of \(G\)-subspaces \(X^n\) of \(X\) such that \(X^0\) is a disjoint union of orbits, \(X^{n+1}\) is obtained from \(X^n\) by attaching \(G\)-cells of the form \(G\times_H DV\), where \(H\) is a closed subgroup of \(G\) and \(DV\) is the unit \(G\)-invariant disk of a finite-dimensional \(G\)-representation space \(V, X =\bigcup X^n\), and \(X\) has the colimit topology derived from the \(G\)-subspaces \(X^n\). When forming \(X^{n+1}\) from \(X^n\), no restriction is imposed on the dimension of \(G\times_H DV\). A \(G\)-cell \(G\times_H DV\) is said to be even-dimensional if the fixed point set \(V^K\) is even-dimensional over \(\mathbb{R}\) for every subgroup \(K\) of \(H\).
The theorem asserts that if \(G = \mathbb{Z}/p\) for a prime \(p\), and \(X\) is a finite \(\text{Rep}^*(G)\)-cell complex formed from only even-dimensional \(G\)-cells, then the \(RO(G)\)-graded Mackey functor-valued equivariant ordinary homology \(H^G_*(X;A)\) of \(X\) with Burnside ring coefficients is free over \(H_*\), the equivariant ordinary \(RO(G)\)-graded homology of a point with Burnside ring coefficients. Moreover, there is a one-to-one correspondence between the generators of \(H^G_*(X;A)\) and the \(G\)-cells of \(X\). However, the dimension of each generator is not necessarily the same as the dimension of the corresponding \(G\)-cell. To prove the theorem, the authors consider the homology of \(X\) with several different choices of coefficients and they apply the Universal Coefficient Theorem for \(RO(G)\)-graded equivariant homology. In order to make use of the Universal Coefficient Theorem, they introduce the box product of \(RO(G)\)-graded Mackey functors, and they also compute the \(RO(G)\)-graded equivariant ordinary homology of a point with an arbitrary Mackey functor as coefficients.