On nonsymmetric theorems for \((H, G)\)-coincidences
This paper presents a generalization of the classical Borsuk-Ulam theorem. Let \(G\) be a finite group acting freely on \(S^n\), and \(H\) a normal cyclic subgroup of \(G\) with prime order. For any map \(f: X\to Y\) and \(\varphi: X\to S^n\), a set \(A_{\varphi}(f, H, G)\) of \((H,G)\)-coincidence points relative to \(\varphi\) is defined, which is a kind of generalization of the usual coincidence set. The main result of this paper shows that the cohomology dimension of the sets \(A_{\varphi}(f, H, G)\) is bounded from below by \(n - |G| \dim Y\) if \(Y\) is a CW-complex, \(\varphi^*: H^n(S^n, \mathbb{Z}_p)\to H^n(X, \mathbb{Z}_p)\) is non-zero, and \(f^*: H^i(Y, \mathbb{Z}_p)\to H^i(X, \mathbb{Z}_p)\) is trivial for \(i\geq 1\). This is a non-symmetric version of the main result in [\textit{D. Gonçalves, J. Jaworowski} and \textit{P. Pergher}, Topology Appl. 125, No.~3, 465--470 (2002; Zbl 1031.55001)].
scientific article; zbMATH DE number 5578032