Discrete Morse theory

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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[edit]

Let \mathcal{X} be a CW complex. Define the incidence function \kappa:\mathcal{X} \times \mathcal{X} \to \mathbb{Z} in the following way: given two cells \sigma and \tau in \mathcal{X}, let \kappa(\sigma,~\tau) be the degree of the attaching map from the boundary of \sigma to \tau. The boundary operator \partial on \mathcal{X} is defined by

\partial(\sigma) = \sum_{\tau \in \mathcal{X}}\kappa(\sigma,\tau)\tau

It is a defining property of boundary operators that \partial\circ\partial \equiv 0. In more axiomatic definitions[5] one can find the requirement that \forall \sigma,\tau^{\prime} \in \mathcal{X}

 \sum_{\tau \in \mathcal{X}} \kappa(\sigma,\tau) \kappa(\tau,\tau^{\prime}) = 0

which is a corollary of the above definition of the boundary operator and the requirement that \partial\circ\partial \equiv 0.

Discrete Morse functions[edit]

A real-valued function \mu:\mathcal{X} \to \mathbb{R} is a discrete Morse function if it satisfies the following two properties:

  1. For any cell \sigma \in \mathcal{X}, the number of cells \tau \in \mathcal{X} in the boundary of \sigma which satisfy \mu(\sigma) \leq \mu(\tau) is at most one.
  2. For any cell \sigma \in \mathcal{X}, the number of cells \tau \in \mathcal{X} containing \sigma in their boundary which satisfy \mu(\sigma) \geq \mu(\tau) 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 \sigma, provided that \mathcal{X} is a regular CW complex. In this case, each cell \sigma \in \mathcal{X} can be paired with at most one exceptional cell \tau \in \mathcal{X}: either a boundary cell with larger \mu value, or a co-boundary cell with smaller \mu 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: \mathcal{X} = \mathcal{A} \sqcup \mathcal{K} \sqcup \mathcal{Q}, where:

  1. \mathcal{A} denotes the critical cells which are unpaired,
  2. \mathcal{K} denotes cells which are paired with boundary cells, and
  3. \mathcal{Q} denotes cells which are paired with co-boundary cells.

By construction, there is a bijection of sets between k-dimensional cells in \mathcal{K} and the (k-1)-dimensional cells in \mathcal{Q}, which can be denoted by p^k:\mathcal{K}^k \to \mathcal{Q}^{k-1} for each natural number k. It is an additional technical requirement that for each K \in \mathcal{K}^k, the degree of the attaching map from the boundary of K to its paired cell p^k(K) \in \mathcal{Q} is a unit in the underlying ring of \mathcal{X}. For instance, over the integers \mathbb{Z}, the only allowed values are \pm 1. This technical requirement is guaranteed when one assumes that \mathcal{X} is a regular CW complex over \mathbb{Z}.

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

The Morse complex[edit]

A gradient path is a sequence of paired cells

\rho = (Q_1, K_1, Q_2, K_2, \ldots, Q_M, K_M)

satisfying Q_m = p(K_m) and \kappa(K_m,~Q_{m+1}) \neq 0. The index of this gradient path is defined to be the integer

\nu(\rho) = \frac{\sum_{m=1}^{M-1}-\kappa(K_m,Q_{m+1})}{\sum_{m=1}^{M}\kappa(K_m,Q_m)}.

The division here makes sense because the incidence between paired cells must be \pm 1. Note that by construction, the values of the discrete Morse function \mu must decrease across \rho. The path \rho is said to connect two critical cells A,A' \in \mathcal{A} if \kappa(A,Q_1) \neq 0 \neq \kappa(K_M,A'). This relationship may be expressed as A \stackrel{\rho}{\to} A'. The multiplicity of this connection is defined to be the integer m(\rho) = \kappa(A,Q_1)\cdot\nu(\rho)\cdot\kappa(K_M,A). Finally, the Morse boundary operator on the critical cells \mathcal{A} is defined by

\Delta(A) = \kappa(A,A') + \sum_{A \stackrel{\rho}{\to} A'}m(\rho) A'

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

Basic Results[edit]

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

The Morse Inequalities[edit]

Let \mathcal{A} be a Morse complex associated to the CW complex \mathcal{X}. The number m_q = |\mathcal{A}_q| of q-cells in \mathcal{A} is called the q^{th} Morse number. Let \beta_q denote the q^{th} Betti number of \mathcal{X}. Then, for any N > 0, the following inequalities[7] hold

m_N \geq \beta_N, and
m_N - m_{N-1} + \ldots \pm m_0 \geq \beta_N - \beta_{N-1} + \ldots \pm \beta_0

Moreover, the Euler characteristic \chi(\mathcal{X}) of \mathcal{X} satisfies

\chi(\mathcal{X}) = m_0 - m_1 + \ldots \pm m_{\dim \mathcal{X}}

Discrete Morse Homology and Homotopy Type[edit]

Let \mathcal{X} be a regular CW complex with boundary operator \partial and a discrete Morse function \mu:\mathcal{X} \to \mathbb{R}. Let \mathcal{A} be the associated Morse complex with Morse boundary operator \Delta. Then, there is an isomorphism[8] of Homology groups as well as homotopy groups.

H_*(\mathcal{X},\partial) \simeq H_*(\mathcal{A},\Delta)

See also[edit]

References[edit]

  1. ^ F. Mori and M. Salvetti: (Discrete) Morse theory for Configuration spaces
  2. ^ Perseus: Software for computing persistent homology groups.
  3. ^ U. Bauer, C. Lange, and M. Wardetzky: Optimal Topological Simplification of Discrete Functions on Surfaces
  4. ^ T Lewiner, H Lopez and G Tavares: Applications of Forman's discrete Morse theory to topological visualization and mesh compression
  5. ^ Mischaikow, Konstantin; Nanda, Vidit. "Morse Theory for Filtrations and Efficient computation of Persistent Homology". Springer. Retrieved 3 August 2013. 
  6. ^ Forman, Robin: Morse Theory for Cell Complexes, Lemma 2.5
  7. ^ Forman, Robin: Morse Theory for Cell Complexes, Corollaries 3.5 and 3.6
  8. ^ Forman, Robin: Morse Theory for Cell Complexes, Theorem 7.3