A Tverberg type theorem for collectively unavoidable complexes

Abstract

We prove that the symmetrized deleted join SymmDelJoin(K) of a “balanced family” K= 〈Ki〉i=1r 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 which is more conceptual and more general than previously known results. Already the case r = 2 of this result 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.