# Questions tagged [discrete-morse-theory]

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

**6**

votes

**0**answers

141 views

### Existence of a perfect discrete Morse function

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 ...

**4**

votes

**0**answers

107 views

### Presentation of the Rybnikov matroid

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 ...

**3**

votes

**0**answers

116 views

### Does a polytope have a self-indexing shelling?

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 ...

**2**

votes

**0**answers

61 views

### Finite cover gives a lift of discrete Morse function

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 ...

**1**

vote

**0**answers

18 views

### Complex of graphs with domination number greater than k

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 ...