Morse matching with 0-cells and (n-1)-cells

Question

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.

Answer 1

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.

Answer 2

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.)