On the existence of \(G\)-equivariant maps
At times, the non-existence of a certain kind of map has interesting consequences, such as the Borsuk--Ulam Theorem. In this paper the authors show the non--existence of certain \(G\)--equivariant maps between free \(G\)--spaces. Let \(R\) be a PID and \(G\) a compact Lie group. Denote the \(i\)--th Betti number of \(X\) by \(\beta_i(X;R)\) and by \(i(X;R)\) the numerical index, cf. \textit{A. Yu. Volovikov} [Sb. Math. 191, No. 9, 1259--1277 (2000); translation from Mat. Sb. 191, No. 9, 3--22 (2000; Zbl 0987.57016)]. Let \(G\) be a compact Lie group and \(X\), \(Y\) free \(G\)--spaces, Hausdorff, pathwise connected and paracompact. Suppose that for some natural number \(m \geq 1\), \(i(X,R)\geq m+1\) and \(H^{k+1}(Y/G;R) = 0\) for some \(k\), \(1\leq k \leq m\). Then if \(k=m\) and \(\beta_m(X;R) < \beta_{m+1}(BG;R)\), there is no \(G\)--equivariant map \(f:X\to Y\). if \(1 \leq k \leq m\) and \(0 < \beta_{k+1}(BG;R)\), there is no \(G\)--equivariant map \(f:X\to Y\).
scientific article; zbMATH DE number 6124359
scientific article