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.

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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 ...
rab's user avatar
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8 votes
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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 ...
Ying Hong Tham's user avatar
9 votes
1 answer
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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 \...
D1811994's user avatar
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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 ...
allizdog's user avatar
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6 votes
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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 ...
Priyavrat Deshpande's user avatar
2 votes
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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 ...
Safia Chettih's user avatar
14 votes
2 answers
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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, $...
Leo's user avatar
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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 ...
snaleimath's user avatar
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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 ...
David Treumann's user avatar
2 votes
1 answer
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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 ...
snaleimath's user avatar
6 votes
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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 ...
Safia Chettih's user avatar
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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 ...
guest's user avatar
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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. ...
Brian Rushton's user avatar
4 votes
2 answers
880 views

Discrete Morse theory and existence of minimal complex

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 ...
Priyavrat Deshpande's user avatar