A Tverberg type theorem for collectively unavoidable complexes

Duško Jojić; Gaiane Panina; Rade Živaljević

paper

A Tverberg type theorem for collectively unavoidable complexes

arXiv:1812.00366

Abstract

We prove (Theorem 2.4) that the symmetrized deleted join $S y mm D e l J o in (K)$ of a "balanced family" $K = ⟨ K_{i} ⟩_{i = 1}^{r}$ of collectively $r$ -unavoidable subcomplexes of $2^{[m]}$ is $(m - r - 1)$ -connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result (Theorem 3.2) which is more conceptual and more general then previously known results. Already the case $r = 2$ of Theorem 3.2 seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman's discrete Morse theory.