Tverberg cores and Kalai's cascade conjecture

Abstract

We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an $n$-simplex to $\mathbb R^d$, let $T_r(f)$ be the set of points contained in the images of $r$ pairwise disjoint faces. We prove that if $r$ is a prime power and $\dim T_r(f)\le k$, then there exists a point that remains an $r$-Tverberg point after any $t$ vertices are deleted, provided $n=(r-1)(d+1)+t(k+1)$. For $t=1$, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is $0$-dimensional.