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?