# Discrete Morse theory

**Discrete Morse theory** is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces,^{[1]} homology computation,^{[2]} denoising,^{[3]} and mesh compression.^{[4]}

## Notation regarding CW complexes

Let be a CW complex. Define the *incidence function* in the following way: given two cells and in , let be the degree of the attaching map from the boundary of to . The boundary operator on is defined by

It is a defining property of boundary operators that . In more axiomatic definitions^{[5]} one can find the requirement that

which is a corollary of the above definition of the boundary operator and the requirement that .

## Discrete Morse functions

A real-valued function is a *discrete Morse function* if it satisfies the following two properties:

- For any cell , the number of cells in the boundary of which satisfy is at most one.
- For any cell , the number of cells containing in their boundary which satisfy is at most one.

It can be shown^{[6]} that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell , provided that is a *regular* CW complex. In this case, each cell can be paired with at most one exceptional cell : either a boundary cell with larger value, or a co-boundary cell with smaller value. The cells which have no pairs, i.e., their function values are strictly higher than their boundary cells **and** strictly lower than their co-boundary cells are called *critical* cells. Thus, a discrete Morse function partitions the CW complex into three distinct cell collections: , where:

- denotes the
**critical**cells which are unpaired, - denotes cells which are paired with boundary cells, and
- denotes cells which are paired with co-boundary cells.

By construction, there is a bijection of sets between -dimensional cells in and the -dimensional cells in , which can be denoted by for each natural number . It is an additional technical requirement that for each , the degree of the attaching map from the boundary of to its paired cell is a unit in the underlying ring of . For instance, over the integers , the only allowed values are . This technical requirement is guaranteed when one assumes that is a regular CW complex over .

The fundamental result of discrete Morse theory establishes that the CW complex is isomorphic on the level of homology to a new complex consisting of only the critical cells. The paired cells in and describe *gradient paths* between adjacent critical cells which can be used to obtain the boundary operator on . Some details of this construction are provided in the next section.

## The Morse complex

A *gradient path* is a sequence of paired cells

satisfying and . The *index* of this gradient path is defined to be the integer

The division here makes sense because the incidence between paired cells must be . Note that by construction, the values of the discrete Morse function must decrease across . The path is said to *connect* two critical cells if . This relationship may be expressed as . The *multiplicity* of this connection is defined to be the integer . Finally, the **Morse boundary operator** on the critical cells is defined by

where the sum is taken over all gradient path connections from to .

## Basic Results

Many of the familiar results from continuous Morse theory apply in the discrete setting.

### The Morse Inequalities

Let be a Morse complex associated to the CW complex . The number of -cells in is called the *Morse number*. Let denote the Betti number of . Then, for any , the following inequalities^{[7]} hold

Moreover, the Euler characteristic of satisfies

### Discrete Morse Homology and Homotopy Type

Let be a regular CW complex with boundary operator and a discrete Morse function . Let be the associated Morse complex with Morse boundary operator . Then, there is an isomorphism^{[8]} of Homology groups as well as homotopy groups.

## See also

- Digital Morse theory
- Stratified Morse theory
- Piece-wise linear Morse theory
- Shape analysis
- Topological combinatorics
- Discrete differential geometry

## References

- ↑ F. Mori and M. Salvetti: (Discrete) Morse theory for Configuration spaces
- ↑ Perseus: the Persistent Homology software.
- ↑ U. Bauer, C. Lange, and M. Wardetzky: Optimal Topological Simplification of Discrete Functions on Surfaces
- ↑ T Lewiner, H Lopez and G Tavares: Applications of Forman's discrete Morse theory to topological visualization and mesh compression
- ↑ Template:Cite web
- ↑ Forman, Robin:
*Morse Theory for Cell Complexes*, Lemma 2.5 - ↑ Forman, Robin:
*Morse Theory for Cell Complexes*, Corollaries 3.5 and 3.6 - ↑ Forman, Robin:
*Morse Theory for Cell Complexes*, Theorem 7.3

- Robin Forman (2002) A User's Guide to Discrete Morse Theory, Séminare Lotharinen de Combinatore 48
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- nLab Article [1]