Generalized chessboard complexes and discrete Morse theory

Abstract

Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a
tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and
combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted
as a result about a critical point of a discrete Morse function on the Bier sphere Bier(K) of an
associated simplicial complex K. We illustrate the use of “standard discrete Morse functions”
on generalized chessboard complexes by proving a connectivity result for chessboard complexes
with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for j-wise disjoint partitions of a simplex.