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Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.

If $P$ is connected and has one maximum element, can I conclude that the simplicial complex associated to $P$ (minus the maximal element) is homotopy equivalent to a disjoint union of spheres?

I know that if one has only one 0-cell (and the some number of $(n-1)$-cells) then the complex would be homotopic to a wedge of spheres, so I was wondering what would happen if one has more than one 0-cell.

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Is the poset assumed connected? –  Benjamin Steinberg Jul 1 '13 at 22:32
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@BenjaminSteinberg how can a connected poset be homotopy equivalent to a disjoint union of spheres unless something very trivial is going on? –  Vidit Nanda Jul 1 '13 at 23:06
    
Do you want $P$ to have exactly $k$ pieces, or is a smaller number acceptable? –  Vidit Nanda Jul 1 '13 at 23:10
    
Couldn't you get a disjoint union of several wedges of spheres and several isolated vertices? –  Dan Ramras Jul 2 '13 at 0:12
    
@DanRamras I guess this is possible. Are there any other possibilities? Due to how the attaching maps work, the intersection of two $(n-1)$-cells can only be empty or one point. –  guest Jul 2 '13 at 1:03

2 Answers 2

Based on the comments under the question, it seems that the real question is as follows:

Let $P$ be a finite poset whose maximal chains have length $\leq n+1$ for some strictly positive $n \in \mathbb{N}$. Assume the existence of a Morse matching which only admits critical cells in dimensions $n$ and $0$. Is it true that the order complex $\Delta(P)$ of our original poset is homotopy-equivalent to a disjoint union of wedges-of-spheres?

The answer is yes, and in fact much more can be concluded.

Let $c_1,\ldots,c_k$ be the critical cells of dimension $n$ and $d_1,\ldots,d_\ell$ be the critical cells of dimension $0$ (one can assume $k=\ell$ if one wishes, but it is completely unnecessary). A gradient path from $c_i$ to $d_j$ is an alternating sequence of elements in $P$ given by

$$ c_i > e_1 < \mu(e_1) > e_2 < \mu(e_2) > \cdots > e_p < \mu(e_p) > d_j$$

where $\mu$ denotes our acyclic matching. It follows by acyclicity of $\mu$ that no such path can be a loop. Note also that there can be no paths between two $c$-cells or two $d$-cells by dimension considerations alone. Associate to each critical cell $c_i$ its "unstable manifold" $W_i^-$ consisting of cells on all paths starting from $c_i$, and associate to each critical cell $d_j$ its "stable manifold" $W_j^+$ consisting of cells on all paths ending at $d_j$. It follows from Forman's first discrete Morse lemma that each $W_i^-$ simple homotopy collapses to $c_i$ and $W_j^+$ simple homtopy collapses to $d_j$, so in fact each stable and unstable manifold in sight is contractible.

It is well-known that the union of two contractible simplicial complexes is homotopy-equivalent to the suspension of their intersection. Applying this rule to each critical cell pair $(d_j,c_i)$ tells you not only that you have a disjoint union of wedges of $n$-spheres, but also which spheres are wedged together along which points.

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The statement in the last paragraph is rather famously false without some extra hypotheses (which are satisfied in this setting) . If you take the circle and write it as a union of two half-open arcs with empty intersection, you get a counterexample. This is the problem that was identified by Mnev in Biss's (faulty) proof of Macpherson's matroid Grassmannian conjecture. Here it's fine because you're talking about simplicial complexes. –  Dan Ramras Jul 2 '13 at 16:47
    
Thanks, Dan! I remember Mnev's correction of Biss's papers, and really should have kept it in mind. –  Vidit Nanda Jul 2 '13 at 16:51
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Thanks for the nice answer, Vidit. It's easy to forget how useful the flow lines really are. Sometimes this gets lost in discussions of Forman-style discrete Morse theory. –  Dan Ramras Jul 2 '13 at 18:45

In this answer, I want to 1) give a shorter approach than the above (avoiding gradient paths) to see that $\Delta P$ is homotopy equivalent to a disjoint union of bouquet of spheres plus isolated vertices, and also 2) answer the reference request from Dan Ramras above (since my references didn't fit into a comment.)

  1. Assuming $d>1$: If a simplicial complex $\Delta$ has a Morse matching with critical cells in dimensions 0 and $d$, then $\Delta$ is homotopy equivalent to a CW-complex with cells corresponding to the critical cells. A CW-complex is built up inductively by gluing $i$ cells along their (entire) boundary to the $i-1$ skeleton (see Hatcher).
    The $d-1$ skeleton consists of a set of disjoint vertices. There is no continuous map from $S^{d-1}$ to $S^0$, so each $d$ cell attaches to a single vertex. It now reduces to the case of a single vertex, where you're exactly attaching some number (possibly 0) of $d$-discs along their entire boundary to a point, giving either a bouquet of $d$-spheres or (in the case of 0 discs) a point.
    (If $d=1$, then one exactly gets a graph, which is also homotopy equivalent to a disjoint union of bouquets of 1-spheres plus isolated vertices by e.g. Hatcher.)

  2. References. The first explicit statement of Morse matchings in terms of acyclic matchings of the face poset that I know of is

    • Manoj Chari, On discrete Morse functions and combinatorial decompositions.

    As a mathematical child of Ken Brown, I also want to note that it's essentially contained (in somewhat greater generality, which muddies the statement somewhat) in the paper

    • Ken Brown, The geometry of rewriting systems -- a proof of the Anick-Groves-Squier theorem.

    Some version of this must also be contained in Forman's articles, but I'm less familiar with these. Certainly Chari felt it to be worthwhile to state it explicitly in the above paper, and Ken mentioned at some point that he didn't see the connection between his paper and Forman's papers before Chari made this observation.

    Discrete Morse theory (at least as introduced by Forman and/or Brown, but see the comment by Vidit Nanda below) relates a simplicial complex with a (simpler) CW-complex via a series of elementary collapses, and in particular doesn't make sense outside of the realm of simplicial complexes. But if you want to work on the order complex of a poset "in terms of the poset", then the right places to start are the articles:

    • Eric Babson and Patricia Hersh, Discrete Morse functions from lexicographic orders and
    • Patricia Hersh, On optimizing discrete Morse functions.

    (These articles give what can be seen as a generalization of EL-shellability to Morse matchings.) A very nice exposition of the poset discrete Morse techniques of Babson and Hersh is given by Sagan and Vatter:

    • Bruce Sagan and Vincent Vatter, The Moebius function of the composition poset.

One more comment:
When I'm explaining discrete Morse Theory shortly to non-geometric combinatorics people, I explain homology as a way to do linear-algebraic matchings in a way that is often helpful to inclusion-exclusion problems. The discrete Morse matchings go back and use plain old matchings to do the linear-algebraic matching. (Of course, discrete Morse matchings also give a stronger homotopy equivalence statement, which is pleasing.)

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Russ, I'm not sure why you say "discrete Morse theory doesn't make sense outside the realm of simplicial complexes". See, for instance, Dmitry Kozlov's disrete Morse theory for free chain complexes, or Skoldberg's discrete Morse theory from an algebraic viewpoint, not to mention two of my own papers. –  Vidit Nanda Jul 5 '13 at 14:43
    
I stand corrected -- there are indeed some other variants of discrete Morse theory that apply to other objects. Maybe there is also one for posets. Certainly a similar matching technique will always help calculate the Moebius number (but this is near trivial.) In any case, I was thinking of the original theory of Forman, Brown, et al. I'll try to edit to reflect. –  Russ Woodroofe Jul 5 '13 at 16:37
    
Russ, even Forman's original paper -- called Morse theory for Cell complexes -- applies to regular CW complexes, which one could argue are more general than simplicial complexes. It's not mysterious: any time the degree of a codimension-1 attaching map is a unit, you can pair the two cells. For regular CW complexes, we always have unit attachment so everything works out "trivially". –  Vidit Nanda Jul 5 '13 at 18:17
    
Vidit, I'm indeed well aware that Forman and Brown's papers both apply to regular CW complexes (and indeed even a bit more broadly). But 1) regular CW-complexes correspond pretty directly with simplicial complexes via barycentric subdivision, and 2) a regular CW-complex is still a geometric object. I was saying, if somewhat imprecisely, that discrete Morse Theory questions are interesting on the order complex of the poset, and not so much on the poset itself. –  Russ Woodroofe Jul 6 '13 at 7:34
    
Russ, thanks for clarifying, I understand what you are trying to say now. In light of Kozlov's book "Combinatorial Algebraic topology" and particularly Chapter 11, called "discrete Morse theory for posets", it seems that discrete Morse theory has been defined directly on posets. But maybe I have misunderstood something. –  Vidit Nanda Jul 6 '13 at 13:57

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