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Discrete Morse Theory is a combinatorial analogue of Morse Theory, introduced by Forman. It provides techniques for computing homological properties of simplicial sets/complexes.

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Any two isomorphic simplicial complexes are simple-homotopy-equivalent. This is a fairly simple result, but it is not obvious. Yet I have been surprisingly unable to find it in the literature on ...
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First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$. For simplicity, let's restrict for now to the ...
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Is there an analog of Cerf theory in PL? More specifically, given two handle decompositions of a PL (relative) cobordism $W$, is it always possible to go from one handle decomposition to the other via ...
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Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \...
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I am studying discrete Morse theory and as an example, discrete Morse theory is used to obtain the homotopy type of the complex of non-connected graphs of $n$ vertices. I also read that this kind of ...
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Let $X$ denote a regular cell structure on a closed (orientable) $n$-manifold (If it helps, the cells are polytopal and the attaching maps are affine). Recall that a discrete Morse function on this ...
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Let's say I have a finite simplicial complex $X$ with a finite covering map $\pi: \widetilde{X} \rightarrow X$ and a discrete gradient vector field $V$ on $X$ (which for my purposes I prefer to its ...
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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, $...
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In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...
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If $X$ is a smooth projective toric variety and $P \subset \mathbf{R}^n$ is its moment polytope, then a generic linear function on $\mathbf{R}^n$ induces (1) a Morse function on $X$, and (2) a ...
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I'm working on some presentation of the fundamental group of the complement of a complex lines arrangement in $\mathbb{C}^{2}.$ In particular, in Arvola's article "The fundamental group of the ...
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In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...
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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 ...
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There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. ...
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A minimal complex is a CW complex whose only cells are the homology cells. Is there some sort of criterion on CW complexes about existence of minimal complexes? Actually I am working on a problem ...
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