Orbit spaces of free involutions on the product of three spheres (Q6572151)

Let \(G=\mathbb{Z}_2\) act on a finitistic space \(X\) (A paracompact Hausdroff space satisfies that every open covering has a finite dimensional open refinement). Associated to the \(G\)-action on \(X\), there is a Borel fibration \(X\rightarrow X_G\rightarrow B_G\), where \(X_G=(X\times E_G)/G\) (Borel space) obtained by diagonal action of \(G\) on space \(X\times E_G\) and \(B_G\) (classifying space) is the orbit space of free action of \(G\) on contractible space \(E_G\) (a infinite join of \(G\) with itself). It is known that the Borel space \(X_G\) is homotopy equivalent to the orbit space \(X/G\).\N\NConsider the free involutions on a finitistic space \(X\), where the finitistic space \(X\) has the mod 2 cohomology of the product of three spheres \(S^n\times S^m\times S^l\), \(1\leq n\leq m\leq l\), the paper under review mainly determined the equivariant cohomology of \(X\) (the chomology of \(X_G\)), i.e. the cohomology algebra of the orbit space \(X/G\).\N\NFinally, as an application, the authors prove two Borsuk-Ulam type results: Let \(G = \mathbb{Z}_2\) act freely on \(X \sim_2 {S}^n \times{S}^m \times {S}^l\), \(1 \leq n \leq m \leq l\).\N\N\textbf{Proposition 4.1.} There does not exist \(G\)-equivariant map from \({S}^d\) to \(X\), for \(d > h\), where \(h\) is one of the following: \(n, m, l, n+m, 2n+l, n+l, m+l\) or \(n+m+l\).\N\N\textbf{Theorem 4.2.} There is no \(G\)-equivariant map \(f : X \to {S}^k\) if \(1 \leq k < i(X)-1\), where \(i(X)\) is one of the following: \(n, m, l, m-n, l-m, l-n\) or \(l-m-n\).