Discrete Morse functions from lexicographic orders

By Eric K. Babson, P. Hersh. 2003

51 citations recorded by the source corpus.

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 O and I 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 Mobius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset Π n /S λ of partitions of a set {1 λ 1,...,k λ k} with repetition is homotopy equivalent to a wedge of spheres of top dimension when A is a hook-shaped partition; it is likely that the proof may be extended to a larger class of A and perhaps to all A, despite a result of Ziegler (1986) which shows that Π n /S λ is not always Cohen-Macaulay.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/e62a718d15c505d9562ede11530618745e93443e

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In Computational Theory and Mathematics

1. Sam Payne
2. Michelle L. Wachs
3. Eric Katz
4. David Jensen
5. Dhruv Ranganathan
6. Matthew Baker
7. John Shareshian
8. Walter Gubler

In Discrete Mathematics and Combinatorics

1. Sam Payne
2. Michelle L. Wachs
3. John Shareshian
4. Eric Katz

Citations in the corpus

• Poset Topology: Tools and Applications (2006, background): “A Morse theory version of lexicographical shellability, which is even more general than Kozlov’s version was formulated by Babson and Hersh [9].”
• A USER'S GUIDE TO DISCRETE MORSE THEORY (2002, unknown): “This theorem was recently put to very good use in [ 2 ], in which discrete Morse”
• Boundary complexes and weight filtrations (2011, background): “le. It is perhaps also worth noting that Babson and Hersh have shown that any shellable pure complex has a discrete Morse ow with one critical vertex and all other critical cells in the top dimension [BH05], so ( n) and Palso have such a ow. 7. Weights from singularities Here we consider jth graded ”