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]}^{[3]} denoising,^{[4]} mesh compression,^{[5]} and topological data analysis ^{[6]}.
Contents
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^{[7]} 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^{[8]} 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., whose 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, for instance, 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^{[9]} hold
- , and
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^{[10]} of homology groups
and similarly for the homotopy groups.
See also
- Digital Morse theory
- Stratified Morse theory
- Shape analysis
- Topological combinatorics
- Discrete differential geometry
References
- ^ F. Mori and M. Salvetti: (Discrete) Morse theory for Configuration spaces Archived April 26, 2012, at the Wayback Machine.
- ^ Perseus: the Persistent Homology software.
- ^ Mischaikow, Konstantin; Nanda, Vidit. "Morse Theory for Filtrations and Efficient computation of Persistent Homology". Springer. Retrieved 3 August 2013.
- ^ 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
- ^ "the Topology ToolKit".
- ^ Mischaikow, Konstantin; Nanda, Vidit. "Morse Theory for Filtrations and Efficient computation of Persistent Homology". Springer. Retrieved 3 August 2013.
- ^ Forman, Robin: Morse Theory for Cell Complexes Archived April 24, 2012, at the Wayback Machine., Lemma 2.5
- ^ Forman, Robin: Morse Theory for Cell Complexes Archived April 24, 2012, at the Wayback Machine., Corollaries 3.5 and 3.6
- ^ Forman, Robin: Morse Theory for Cell Complexes Archived April 24, 2012, at the Wayback Machine., Theorem 7.3
- Robin Forman (2002) A User's Guide to Discrete Morse Theory, Séminare Lotharinen de Combinatore 48
- Dmitry Kozlov (2007). Combinatorial Algebraic Topology. Springer. ISBN 978-3540719618.
- Jakob Jonsson (2007). Simplicial Complexes of Graphs. Springer. ISBN 978-3540758587.
- Peter Orlik, Volkmar Welker (2007). Algebraic Combinatorics: Lectures at a Summer School In Nordfjordeid. Springer. ISBN 978-3540683759.
- nLab Article [1]