On the Topological Tverberg Theorem

We introduce a new ``Winding Number Conjecture'' about maps from the $(d-1)$-skeleton of the $((d+1)(q-1))$-simplex into $\real^d$. This conjecture is equivalent to the Topological Tverberg Theorem. Furthermore, many statements about the Topological Tverberg Theorem transfer to the Winding Number Conjecture, for example all currently proven cases of the Topological Tverberg Theorem as well as Sierksma's conjecture about the number of Tverberg partitions. In the case $d=2$, the Winding Number Conjecture is a statement about complete graphs: It claims that in every image of $K_{3(q-1)+1}$ in the plane either $q-1$ triangles wind around one vertex or $q-2$ triangles wind around the intersection of two edges, where the triangles, edges and vertices are disjoint. We examine which other graphs have this property and find the minimal subgraph of $K_7$ having this property.