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.
16
questions
1
vote
0
answers
47
views
Critical simplices of a discrete gradient vector field
I am currently reading a paper on topological data analysis which includes som discrete Morse theory arguments. I got stuck on a corollary that in the paper I'm reading is described as simply ...
- 11
2
votes
0
answers
160
views
Isomorphic simplicial complexes are simple-homotopy-equivalent: reference?
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 ...
- 33.4k
3
votes
1
answer
229
views
Handle attachment information from Morse function and triangulation
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 ...
- 159
8
votes
1
answer
173
views
Analog of Cerf theory in PL
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 ...
- 186
9
votes
1
answer
263
views
Smooth Morse function from Forman's discrete Morse function
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 \...
- 909
1
vote
0
answers
27
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 ...
- 111
6
votes
0
answers
260
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 ...
- 1,017
2
votes
0
answers
86
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 ...
14
votes
2
answers
1k
views
Discrete Morse theory: how do zig-zag paths determine homotopy type?
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, $...
- 1,569
5
votes
1
answer
265
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 ...
- 741
3
votes
0
answers
145
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 ...
- 4,859
2
votes
1
answer
168
views
Complex lines arrangements from a given wiring diagram
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 ...
- 741
6
votes
2
answers
507
views
Using Discrete Morse Theory to represent hom classes
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 ...
0
votes
2
answers
535
views
Morse matching with 0-cells and (n-1)-cells
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 ...
- 9
20
votes
1
answer
902
views
Discrete Morse theory and chess
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. ...
- 3,309