Discrete Morse functions from lexicographic orders

Abstract

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 with0 and1 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 Πn/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 λ is a hook-shaped partition; it is likely that the proof may be extended to a larger class of λ and perhaps to all λ, despite a result of Ziegler (1986) which shows that Πn/S λ is not always Cohen-Macaulay.

Poset topology: Tools and applications

“…A relative version of lexicographic shellability was introduced by Stanley [170]. A Morse theory version of lexicographical shellability, which is even more general than Kozlov's version was formulated by Babson and Hersh [9].…”

The poset of bipartitions

“…In this paper we will use a method developed by Babson and Hersh [1] , in the extended form of Hersh and Welker [15] (which incorporates a correction to [1] pointed out in [14] , [20] ). This method is designed to find the homotopy type of the order complex of a graded partially ordered set with minimum element and maximum element .…”

“…Recall that the order complex of a partially ordered set is the simplicial complex whose vertices are the elements of and whose faces are the chains of . Babson and Hersh [1] find a Morse matching on the Hasse diagram of the poset of faces of , the order relation being defined by inclusion, by fixing an enumeration of the maximal chains of , which they call poset lexicographic order . It was observed by Hersh and Welker [15, Theorem 3.1] that the key property of a poset lexicographic order that is used in all proofs of Babson and Hersh in [1] is that the enumeration of maximal chains considered grows by creating skipped intervals (which is implicit in [1, Remark 2.1] ).…”

“…The proofs of these facts form the most difficult part of our paper. They are essentially based on an adaptation of the Gray code of permutations due to Johnson [16] and Trotter [23] and on work of Babson and Hersh [1] (in the extended form given by Hersh and Welker [15] ) constructing a discrete Morse function in the sense of Forman [8] , [9] , [10] for the order complex of a graded poset. The former is needed to decompose into a union of distributive lattices in a shelling-like manner.…”

“…This decomposition is then refined using the well-known -shelling of distributive lattices in order to obtain an enumeration of the maximal chains of to which the results of Babson, Hersh, and Welker apply. (As Example 9.1 shows, our enumeration of the maximal chains of is not a poset lexicographic order in the sense of [1] , so we do indeed need the extended form observed in [15] .) We remark that our “two-phase” decomposition is similar in spirit to constructions by Hanlon et al.…”