\((H,G)\)-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number \(r\) (Q1704371)

Let \(X\) be a paracompact space, \(G\) a finite group acting freely on \(X\) and \(H\) a cyclic subgroup of \(G\) of prime order \(p\). Consider a continuous map \(f : X \to M\), where \(M\) is a connected \(m\)-manifold (orientable if \(p >2\)) and \(f_\ast(V_k) = 0\) for \(k\geq 1\), where \(V_k\) are the Wu classes of \(M\). Suppose that \(\text{ind }X>n >(|G|-r)m\), where \(r = \frac{|G|}{p}\). The authors estimate the cohomological dimension of the set \(A( f,H,G)\) of \((H, G)\)-coincidence points (defined by \textit{D. L. Gonçalves} and \textit{P. L. Q. Pergher} [Kobe J. Math. 15, No. 2, 191--195 (1998; Zbl 0941.55004)]) of the map \(f\), being a version of the Borsuk-Ulam theorem and a version for manifolds of the authors' previous result. Further, the index of the \((H,G)\)-coincidence set in the case that \(H\) is a \(p\)-torus subgroup of a particular group \(G\) is studied. As an application, a topological Tverberg type theorem for any natural number \(r\) is proved. Such a result is a weak version of the famous topological Tverberg conjecture, which was shown recently to fail for all \(r\) that are not prime powers. At the end, a generalized van Kampen-Flores type theorem for any integer \(r\) is presented.