Let $\Delta$ be a simplicial complex (or more generally, a regular CW complex). Let $\mathcal{M}$ be a Morse matching (or equivalently, a discrete Morse function) on $\Delta$.
By Forman's theorems, $\Delta$ is homotopy equivalent to a CW-complex whose cells are $\mathcal{M}$-critical (=unmatched) simplices. Its homology is computed from the chain complex, in which an entry of the $k$-th boundary matrix equals the sum of signs of all zig-zag paths (=directed paths in the Hasse diagram of $\Delta$ with the edges from $\mathcal{M}$ reversed) between a critical $k$-cell and critical $k\!-\!1$-cell.
Do zig-zag paths determine how the critical cells are glued onto each
other?
A CW-complex $X$ is determined, up to homotopy, by the number of cells in each dimension, and the homotopy class of each gluing map $S^k \!\longrightarrow\! X^{(k)}$.
Q1: If all critical cells are in dimensions $0,k,k\!+\!1$, then gluing
maps are determined by equivalence classes in
$\pi_k(S^k)\cong\mathbb{Z}$. Does the sum of the signs of all zig-zag
paths from a $k\!+\!1$-cell to a $k$-cell equal the degree of the
gluing map?
I suspect this to be true. More intriguingly:
Q2: If all critical cells are in dimensions $0,k,k\!+\!2$ with $k\!\geq\!3$, then gluing
maps are determined by equivalence classes in
$\pi_{k+1}(S^k)\cong\mathbb{Z}_2$. Does the sum of the signs of all paths (of a new type) from a $k\!+\!2$-cell to a $k$-cell determine the gluing map?
In general, the homotopy groups of spheres are not all of the form $\mathbb{Z}_m$, so probably summing the signs of zig-zag paths is not sufficient, i.e. some information is lost when applying the matching. For instance, if critical cells are of dimension $1,4,8$, then gluing maps are determined by equivalence classes in $\pi_7(S^4)\cong\mathbb{Z}\!\oplus\!\mathbb{Z}_{12}$. Furthermore, the gluing maps of $k\!+\!1$-cells go into the $k$-skeleton, so they are determined by their representatives in $\pi_k(X^{(k)})$, but according to Mihai Damian, On the higher homotopy groups of a finite CW-complex, Topology Appl. 149 (2005), no. 1-3, 273–284., a homotopy group of a finite CW-complex may be infinitely-generated!
I have an example of $\Delta$ and $\mathcal{M}$ that produce critical $1$ $0$-simplex, $1$ $5$-simplex, $10$ $7$-simplices. Can I conclude that $\Delta\simeq S^5\vee\bigvee_{\!10}S^7$ by inspecting certain (???) paths in the Hasse diagram?
Best Answer
Thanks to Cosheaf Overlord Justin Curry for bringing this question to my attention. I'm only going to address the first question here, and I think with some computations (whose complexity depends on your individual complexes) you can extract answers to the other two questions yourself.
There is a very precise relation between zigzags of an acyclic partial matching $\mathcal{M}$ and the homotopy type of $\Delta$, namely:
All details are in the preprint here; the category mentioned above is a discrete analogue of the flow category described by Cohen, Jones and Segal in this paper.
Here is a brief summary of how to prove the result: given a regular CW complex $\Delta$ one has the "entrance path category" $E_\Delta$ whose objects are the cells and morphisms $E_\Delta(x,y)$ are descending sequences of faces $(x > z_0 > \cdots > z_n > y)$. You compose by concatenation, and note that these descending sequences have a poset structure (by inclusion of subsequences) where $(x > y)$ is minimal in $E_\Delta(x,y)$ if $x \neq y$ and $(x)$, which serves as the identity, happens to be unique and therefore minimal in $E_\Delta(x,x)$. Regularity guarantees that the classifying space of $E_\Delta$ lies in the homotopy class of $\Delta$.
Now, every acyclic partial matching picks out a class of minimal morphisms $\{(x_\bullet > y_\bullet)\}$ which I'll call $\mathcal{M}$. And the zigzags you see are precisely the sorts of morphisms which arise when you localize about (i.e., formally invert) elements of $\mathcal{M}$ in the poset-enriched category $E_\Delta$. Call this localized poset-enriched category $E_\Delta[\mathcal{M}^{-1}]$ and consider its full subcategory $C_\Delta(\mathcal{M})$ whose objects are the critical cells. Here's the main result of that preprint:
The proofs of both equivalences require investigating fiber 2-categories and the appropriate use of Quillen's Theorem A. Things get slightly technical, but you can find an explicit computation in Sec 7.1 of the preprint. In any case, one immediate corollary is the quoted theorem above.