The genus and the category of configuration spaces
If \(X\) is a topological space, we write \(H^*(X,M) = H^*(X\times EG/G;M)\) for a \(\mathbb Z[G]\)-module \(M\) (action by the fundamental group). \(H^*_G(M) = H^*_G(pt;M)\); the map \(X\to pt\) induces \(H^*_G(M)\to H^*_G(X,M)\). The upper cohomological index \(\overline{\text{ind}}_M X\) is the largest \(n\) for which this map is non-trivial (which may be \(\infty\)). A more complex numerical index is \(i_G(X)\), defined by \textit{A. Yu. Volovikov} [Mat. Sb. 191, No.~9, 3--22 (2000; Zbl 0987.57016)]. Let \(G = (\mathbb Z_p)^k\) be the elementary abelian group and \(X\) be a \(G\)-space. Suppose \(H^m(G,\mathbb Z_p), m<n\) are finite \(\mathbb Z_p[G]\) modules which are induced from proper subgroups of \(G\). Then \(i_G(X)\geq n+1\). This is applied to configuration spaces. The author also studies \(G\)-category of \(X\) [see \textit{T. Bartsch}, Topological methods for variational problems with symmetries. Lecture Notes in Mathematics. 1560. Berlin: Springer-Verlag (1993; Zbl 0789.58001)]. The \(G\)-category is estimated from below in terms of the distinct sets of orbits. This is applied to configuration spaces built from points in a manifold.
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