Discrete Morse functions from lexicographic orders

Authors:
Eric Babson and Patricia Hersh

Journal:
Trans. Amer. Math. Soc. **357** (2005), 509-534

MSC (2000):
Primary 05E25; Secondary 05A17, 05A18, 55P15

DOI:
https://doi.org/10.1090/S0002-9947-04-03495-6

Published electronically:
September 2, 2004

MathSciNet review:
2095621

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with and from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Möbius function may be computed as the alternating sum of Morse numbers.

The above construction enables us to prove that the poset of partitions of a set with repetition is homotopy equivalent to a wedge of spheres of top dimension when is a hook-shaped partition; it is likely that the proof may be extended to a larger class of and perhaps to all , despite a result of Ziegler (1986) which shows that is not always Cohen-Macaulay.

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Additional Information

**Eric Babson**

Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Email:
babson@math.washington.edu

**Patricia Hersh**

Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Address at time of publication:
The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720-5070

Email:
phersh@msri.org

DOI:
https://doi.org/10.1090/S0002-9947-04-03495-6

Keywords:
Discrete Morse theory,
poset,
order complex,
partition

Received by editor(s):
July 1, 2003

Published electronically:
September 2, 2004

Article copyright:
© Copyright 2004
by Eric Babson and Patricia Hersh