A topological central point theorem (Q409533)

The author obtains first the following generalized version of the topological Tverberg theorem for maps of a simplex to finite-dimensional metric spaces:NEWLINENEWLINELet \(d \geq 1\), \(r \geq 2\) be integers, \(m = (d+1)(r-1)\), \(\Delta^m\) be the \(m\)-dimensional simplex, and \(W\) be a \(d\)-dimensional metric space. Then for every continuous map \(f: \Delta^m \rightarrow W\), NEWLINE\[NEWLINE \bigcap \{f(F): F\subset \Delta^m,\;\dim F = d(r-1)\} \neq \emptyset.NEWLINE\]NEWLINENEWLINENEWLINEIt is natural to ask whether a similar version of the Tverberg theorem holds for maps from \(\Delta^m\) to a \(d\)-dimensional metric space, at least for \(r\) a prime power. The following positive result is established by the author, increasing the number \(m = (d+1)(r-1)\):NEWLINENEWLINELet \(r\) be a prime power and let \(m = (d+1)(r-1)\). For every continuous map \(f: \Delta^m \rightarrow W\) to a \(d\)-dimensional metric space \(W\), there exist \(r\) disjoint faces of \(\Delta^m\) whose images under \(f\) have a point in common.