Winding Number Conjecture

Let $d$ and $q$ be positive integers. For a continuous map $f:S^{d-1}\to\mathbb{R}^d$ and a point $p\notin f(S^{d-1})$, define the winding number $W(f,p)\in\mathbb{Z}$ by the homology class of $f$ in $\widetilde H_{d-1}(\mathbb{R}^d\setminus\{p\};\mathbb{Z})\cong\mathbb{Z}$. For each $d$-simplex, identify its boundary with $S^{d-1}$ and use the same definition. Winding Number Conjecture. For every continuous map

$$f:\Delta_{(d+1)(q-1)}^{\le d-1}\longrightarrow\mathbb{R}^d$$

there are $q$ disjoint faces $\sigma_1,\dots,\sigma_q$ of $\Delta_{(d+1)(q-1)}^{\le d}$ and a point $p\in\mathbb{R}^d$ such that, for each $i$, either $\dim(\sigma_i)\le d-1$ and $p\in f(\sigma_i)$, or $\dim(\sigma_i)=d$ and either $p\in f(\partial\sigma_i)$ or $p\notin f(\partial\sigma_i)$ and $W(f|_{\partial\sigma_i},p)\ne0$. The conjecture is known in dimension $d=1$, and the paper proves that for each $q\ge2$ it is equivalent to the Topological Tverberg theorem; its general validity remains unresolved.