On a topological generalization of the Tverberg theorem (Q1358498)

Let \(\Delta^N\) denote an \(N\)-dimensional simplex and \(M^m\) an \(m\)-dimensional topological manifold. The main result. Let \(q=p^n\) where \(p\) is an arbitrary prime number, \(N=(q-1)(m+1)\) and \(f:\partial\Delta^N\to M^m\) be a continuous map. For \(q=2\) it is supposed additionally that the modulo 2 degree of the map \(f\) equals zero. Then there exist \(q\) mutually disjoint closed faces of \(\Delta^N\) such that the intersection of their \(f\)-images is nonempty.