The Dowker theorem via discrete Morse theory

The Dowker theorem via discrete Morse theory

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 resul...

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Bibliographic Details
Main Authors: Brun, Morten, Grinberg, Darij
Format: Journal Article
Language:English
Published: 22-07-2024
Subjects:
Mathematics - Algebraic Topology
Mathematics - Combinatorics
Online Access:Get full text
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Summary: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.
DOI:10.48550/arxiv.2407.15454