Borsuk-Ulam theorem for filtered spaces (Q2126291)

Considering a wider class of topological spaces \(X\) which are not necessarily cohomological \(n\)-acyclic spaces, the authors prove that there is no equivariant map \(f:(X,T)\rightarrow (Y,S)\). \textbf{Theorem}: Let \(X\) and \(Y\) be pathwise connected and paracompact Hausdorff spaces equipped with free involutions \(T:X\rightarrow X\) and \(S:Y\rightarrow Y\), respectively. Let us suppose that there exists a sequence \[ (X_i, T_i)\xrightarrow{h_i} (X_{i+1},T_{i+1}) \mbox{ for }1\leq i\leq k, \] where, for each \(i\), \(X_i\) is a pathwise connected and paracompact Hausdorff space equipped with a free involution \(T_i:X_i\rightarrow X_i\) such that \(X_{k+1}=X\) and \(h_i\) is an equivariant map. Assume that for some sequence of natural numbers \(n_0=0<n_1\leq n_2\leq \dots\leq n_k\) the following holds: \begin{itemize} \item[(i)] \(\check{H}^r(X_i;\mathbb{Z}_2)=0\), for \(n_{i-1}<r<n_i\), \(1\leq i\leq k\), \item[(ii)] \(h_i^*:\check{H}^{n_i}(X_{i+1}l\mathbb{Z}_2)\rightarrow \check{H}^{n_i}(X_{i}l\mathbb{Z}_2)\) is the null homomorphism, for \(1\leq i\leq k\), \item[(iii)] \(\check{H}^{n_k+1}(Y/S;\mathbb{Z}_2)=0\), \end{itemize} where \(\check{H}\) denotes Čech cohomology. Then there is no equivariant map \(f:(X,T)\rightarrow (Y,s)\). Let \(h_i\) be the inclusion map, then the authors obtain the following \textbf{Corollary}: Let \(X\) and \(Y\) be pathwise connected and paracompact Hausdorff spaces, equipped with free involutions \(T:X\rightarrow X\) and \(S:Y\rightarrow Y\), respectively. Let us consider \(X\) a filtered space with filtration \[ A_i\subset A_2\subset A_3\subset \cdots\subset A_k=X \] where each element of the filtration is a pathwise connected and paracompact Hausdorff space. Suppose that for some sequence of natural numbers \(n_0=0<n_1\leq n_2\leq \cdots\leq n_k\) the following holds: \begin{itemize} \item[(i)] \(\check{H}^r(A_i;\mathbb{Z}_2)=0\), for \(n_{i-1}<r<n_i\), \(1\leq i\leq k\), \item[(ii)] \(h_i^*:\check{H}^{n_i}(A_{i+1}l\mathbb{Z}_2)\rightarrow \check{H}^{n_i}(A_{i}l\mathbb{Z}_2)\) is the null homomorphism, for \(1\leq i\leq k\), where \(j_i: A_i\hookrightarrow A_{i+1}\) is the natural inclusion, \item[(iii)] \(\check{H}^{n_k+1}(Y/S;\mathbb{Z}_2)=0\), \end{itemize} where \(\check{H}\) denotes Čech cohomology. Then there is no equivariant map \(f:(X,T)\rightarrow (Y,s)\). At the end of the paper, the authors also present some interesting examples to illustrate their results.