The Dowker theorem via discrete Morse theory

TL;DR

This paper provides a new combinatorial proof of Barmak's strengthening of the Dowker theorem, showing that the associated complexes are simple-homotopy equivalent using discrete Morse theory.

Contribution

It introduces an explicit combinatorial construction of acyclic matchings to prove simple-homotopy equivalence in the Dowker complexes.

Findings

Established a combinatorial proof of simple-homotopy equivalence

Constructed explicit acyclic matchings in discrete Morse theory

Reinforced the connection between topology and combinatorics in finite spaces

Abstract

The Dowker theorem is a classical result in the topology of finite spaces, claiming that any binary relation between two finite spaces defines two homotopy-equivalent complexes (the Dowker complexes). Recently, Barmak strengthened this to a simple-homotopy-equivalence. We reprove Barmak's result using a combinatorial argument that constructs an explicit acyclic matching in the sense of discrete Morse theory.