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'''Discrete Morse theory''' is a [[combinatorial]] adaptation of [[Morse theory]] developed by [http://math.rice.edu/~forman/ Robin Forman]. The theory has various practical applications in diverse fields of [[applied mathematics]] and [[computer science]], such as [[configuration space]]s,<ref>F. Mori and M. Salvetti: [http://mrlonline.org/mrl/2011-018-001/2011-018-001-004.pdf (Discrete) Morse theory for Configuration spaces]</ref> [[Homology (mathematics)|homology]] computation<ref>[http://www.sas.upenn.edu/~vnanda/perseus/index.html Perseus]: Software for computing persistent homology groups.</ref> and [[Lossless data compression|mesh compression]].<ref>T Lewiner, H Lopez and G Tavares: [http://www.matmidia.mat.puc-rio.br/tomlew/pdfs/morse_apps_tvcg.pdf Applications of Forman's discrete Morse theory to topological visualization and mesh compression]</ref>
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==Notation regarding CW complexes==
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Let <math>\mathcal{X}</math> be a [[CW complex]]. Define the ''incidence function'' <math>\kappa:\mathcal{X} \times \mathcal{X} \to \mathbb{Z}</math> in the following way: given two cells <math>\sigma</math> and <math>\tau</math> in <math>\mathcal{X}</math>, let <math>\kappa(\sigma,~\tau)</math> be the [[Topological degree theory|degree]] of the [[attaching map]] from the boundary of <math>\sigma</math> to <math>\tau</math>. The [[boundary operator]] <math>\partial</math> on <math>\mathcal{X}</math> is defined by
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:<math>\partial(\sigma) = \sum_{\tau \in \mathcal{X}}\kappa(\sigma,\tau)\tau</math>  
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It is a defining property of boundary operators that <math>\partial\circ\partial \equiv 0</math>. In more axiomatic definitions<ref>{{cite web|last1=Mischaikow|first1=Konstantin|last2=Nanda|first2=Vidit|title=Morse Theory for Filtrations and Efficient computation of Persistent Homology|url=http://link.springer.com/article/10.1007%2Fs00454-013-9529-6|publisher=Springer|accessdate=3 August 2013}}</ref> one can find the requirement that <math>\forall \sigma,\tau^{\prime} \in \mathcal{X}</math>
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:<math> \sum_{\tau \in \mathcal{X}} \kappa(\sigma,\tau) \kappa(\tau,\tau^{\prime}) = 0</math>
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which is a corollary of the above definition of the boundary operator and the requirement that <math>\partial\circ\partial \equiv 0</math>.
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==Discrete Morse functions==
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A [[Real number|real]]-valued function <math>\mu:\mathcal{X} \to \mathbb{R}</math> is a ''discrete Morse function'' if it satisfies the following two properties:
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# For any cell <math>\sigma \in \mathcal{X}</math>, the number of cells <math>\tau \in \mathcal{X}</math> in the boundary of <math>\sigma</math> which satisfy <math>\mu(\sigma) \leq \mu(\tau)</math> is at most one.
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# For any cell <math>\sigma \in \mathcal{X}</math>, the number of cells <math>\tau \in \mathcal{X}</math> containing <math>\sigma</math> in their boundary which satisfy <math>\mu(\sigma) \geq \mu(\tau)</math> is at most one.
 
 
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It can be shown<ref>Forman, Robin: [https://drona.csa.iisc.ernet.in/~vijayn/courses/TopoForVis/papers/FormanDiscreteMorseTheory.pdf ''Morse Theory for Cell Complexes''], Lemma 2.5</ref> that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell <math>\sigma</math>, provided that <math>\mathcal{X}</math> is a ''regular'' CW complex. In this case, each cell <math>\sigma \in \mathcal{X}</math> can be paired with at most one exceptional cell <math>\tau \in \mathcal{X}</math>: either a boundary cell with larger <math>\mu</math> value, or a co-boundary cell with smaller <math>\mu</math> 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: <math>\mathcal{X} = \mathcal{A} \sqcup \mathcal{K} \sqcup \mathcal{Q}</math>, where:
 
 
</ul>
# <math>\mathcal{A}</math> denotes the '''critical''' cells which are unpaired,
# <math>\mathcal{K}</math> denotes cells which are paired with boundary cells, and
# <math>\mathcal{Q}</math> denotes cells which are paired with co-boundary cells.
 
By construction, there is a [[bijection]] of [[Set (mathematics)|sets]] between <math>k</math>-dimensional cells in <math>\mathcal{K}</math> and the <math>(k-1)</math>-dimensional cells in <math>\mathcal{Q}</math>, which can be denoted by <math>p^k:\mathcal{K}^k \to \mathcal{Q}^{k-1}</math> for each [[natural number]] <math>k</math>. It is an additional technical requirement that for each <math>K \in \mathcal{K}^k</math>, the degree of the attaching map from the boundary of <math>K</math> to its paired cell <math>p^k(K) \in \mathcal{Q}</math> is a [[Unit (ring theory)|unit]] in the underlying [[Ring (mathematics)|ring]] of <math>\mathcal{X}</math>. For instance, over the [[integer]]s <math>\mathbb{Z}</math>, the only allowed values are <math>\pm 1</math>. This technical requirement is guaranteed when one assumes that <math>\mathcal{X}</math> is a regular CW complex over <math>\mathbb{Z}</math>.
 
The fundamental result of discrete Morse theory establishes that the CW complex <math>\mathcal{X}</math> is [[isomorphism|isomorphic]] on the level of [[Homology (mathematics)|homology]] to a new complex <math>\mathcal{A}</math> consisting of only the critical cells. The paired cells in <math>\mathcal{K}</math> and <math>\mathcal{Q}</math> describe ''gradient paths'' between adjacent critical cells which can be used to obtain the boundary operator on <math>\mathcal{A}</math>. Some details of this construction are provided in the next section.
 
==The Morse complex==
 
A ''gradient path'' is a sequence of paired cells
 
:<math>\rho = (Q_1, K_1, Q_2, K_2, \ldots, Q_M, K_M)</math>
 
satisfying <math>Q_m = p(K_m)</math> and <math>\kappa(K_m,~Q_{m+1}) \neq 0</math>. The ''index'' of this gradient path is defined to be the integer
 
:<math>\nu(\rho) = \frac{\sum_{m=1}^{M-1}-\kappa(K_m,Q_{m+1})}{\sum_{m=1}^{M}\kappa(K_m,Q_m)}</math>.
 
The division here makes sense because the incidence between paired cells must be <math>\pm 1</math>. Note that by construction, the values of the discrete Morse function <math>\mu</math> must decrease across <math>\rho</math>. The path <math>\rho</math> is said to ''connect'' two critical cells <math>A,A' \in \mathcal{A}</math> if <math>\kappa(A,Q_1) \neq 0 \neq \kappa(K_M,A')</math>. This relationship may be expressed as <math>A \stackrel{\rho}{\to} A'</math>. The ''multiplicity'' of this connection is defined to be the integer <math>m(\rho) = \kappa(A,Q_1)\cdot\nu(\rho)\cdot\kappa(K_M,A)</math>. Finally, the '''Morse boundary operator''' on the critical cells <math>\mathcal{A}</math> is defined by  
 
:<math>\Delta(A) = \kappa(A,A') + \sum_{A \stackrel{\rho}{\to} A'}m(\rho) A'</math>
 
where the sum is taken over all gradient path connections from <math>A</math> to <math>A'</math>.
 
==Basic Results==
 
Many of the familiar results from continuous Morse theory apply in the discrete setting.
 
===The Morse Inequalities===
 
Let <math>\mathcal{A}</math> be a Morse complex associated to the CW complex <math>\mathcal{X}</math>. The number <math>m_q = |\mathcal{A}_q|</math> of <math>q</math>-cells in <math>\mathcal{A}</math> is called the <math>q^{th}</math> ''Morse number''. Let <math>\beta_q</math> denote the <math>q^{th}</math> [[Betti number]] of <math>\mathcal{X}</math>. Then, for any <math>N > 0</math>, the following inequalities<ref>Forman, Robin: [https://drona.csa.iisc.ernet.in/~vijayn/courses/TopoForVis/papers/FormanDiscreteMorseTheory.pdf ''Morse Theory for Cell Complexes''], Corollaries 3.5 and 3.6</ref> hold
 
:<math>m_N \geq \beta_N</math>, and
:<math>m_N - m_{N-1} + \ldots \pm m_0 \geq \beta_N - \beta_{N-1} + \ldots \pm \beta_0</math>
 
Moreover, the [[Euler characteristic]] <math>\chi(\mathcal{X})</math> of <math>\mathcal{X}</math> satisfies
 
:<math>\chi(\mathcal{X}) = m_0 - m_1 + \ldots \pm m_{\dim \mathcal{X}}</math>
 
===Discrete Morse Homology and Homotopy Type===
 
Let <math>\mathcal{X}</math> be a regular CW complex with boundary operator <math>\partial</math> and a discrete Morse function <math>\mu:\mathcal{X} \to \mathbb{R}</math>. Let <math>\mathcal{A}</math> be the associated Morse complex with Morse boundary operator <math>\Delta</math>. Then, there is an [[Group isomorphism|isomorphism]]<ref>Forman, Robin: [https://drona.csa.iisc.ernet.in/~vijayn/courses/TopoForVis/papers/FormanDiscreteMorseTheory.pdf ''Morse Theory for Cell Complexes''], Theorem 7.3</ref> of [[Homology (mathematics)|Homology]] groups as well as homotopy groups. 
 
:<math>H_*(\mathcal{X},\partial) \simeq H_*(\mathcal{A},\Delta)</math>
 
==See also==
*[[Digital Morse theory]]
*[[Stratified Morse theory]]
*[[Piece-wise linear Morse theory]]
*[[Shape analysis]]
*[[Topological combinatorics]]
*[[Discrete differential geometry]]
 
==References==
{{Reflist}}
* Robin Forman (2002) [http://www.emis.de/journals/SLC/wpapers/s48forman.pdf A User's Guide to Discrete Morse Theory], Séminare Lotharinen de Combinatore 48
* {{cite book | author=Dmitry Kozlov | title=Combinatorial Algebraic Topology | publisher=Springer | year=2007 | isbn=978-3540719618}}
* {{cite book | author=Jakob Jonsson | title=Simplicial Complexes of Graphs | publisher=Springer | year=2007 | isbn=978-3540758587}}
* {{cite book | author=Peter Orlik, Volkmar Welker | title=Algebraic Combinatorics: Lectures at a Summer School In Nordfjordeid | publisher=Springer | year=2007 | isbn=978-3540683759}}
 
[[Category:Combinatorics]]
[[Category:Morse theory]]
[[Category:Computational topology]]

Latest revision as of 09:41, 23 December 2014

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