  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.
