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]

Notation regarding CW complexes

Let ${\displaystyle X}$ be a CW complex and denote by ${\displaystyle {\mathcal {X))}$ its set of cells. Define the incidence function ${\displaystyle \kappa \colon {\mathcal {X))\times {\mathcal {X))\to \mathbb {Z} }$ in the following way: given two cells ${\displaystyle \sigma }$ and ${\displaystyle \tau }$ in ${\displaystyle {\mathcal {X))}$, let ${\displaystyle \kappa (\sigma ,~\tau )}$ be the degree of the attaching map from the boundary of ${\displaystyle \sigma }$ to ${\displaystyle \tau }$. The boundary operator is the endomorphism ${\displaystyle \partial }$ of the free abelian group generated by ${\displaystyle {\mathcal {X))}$ defined by

${\displaystyle \partial (\sigma )=\sum _{\tau \in {\mathcal {X))}\kappa (\sigma ,\tau )\tau .}$

It is a defining property of boundary operators that ${\displaystyle \partial \circ \partial \equiv 0}$. In more axiomatic definitions^[7] one can find the requirement that ${\displaystyle \forall \sigma ,\tau ^{\prime }\in {\mathcal {X))}$

${\displaystyle \sum _{\tau \in {\mathcal {X))}\kappa (\sigma ,\tau )\kappa (\tau ,\tau ^{\prime })=0}$

which is a consequence of the above definition of the boundary operator and the requirement that ${\displaystyle \partial \circ \partial \equiv 0}$.

Discrete Morse functions

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

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

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

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

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

The Morse complex

A gradient path is a sequence of paired cells

${\displaystyle \rho =(Q_{1},K_{1},Q_{2},K_{2},\ldots ,Q_{M},K_{M})}$

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

${\displaystyle \nu (\rho )={\frac {\prod _{m=1}^{M-1}-\kappa (K_{m},Q_{m+1})}{\prod _{m=1}^{M}\kappa (K_{m},Q_{m}))).}$

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

${\displaystyle \Delta (A)=\kappa (A,A')+\sum _{A{\stackrel {\rho }{\to ))A'}m(\rho )A'}$

where the sum is taken over all gradient path connections from ${\displaystyle A}$ to ${\displaystyle A'}$.

Basic Results

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

The Morse Inequalities

Let ${\displaystyle {\mathcal {A))}$ be a Morse complex associated to the CW complex ${\displaystyle {\mathcal {X))}$. The number ${\displaystyle m_{q}=|{\mathcal {A))_{q}|}$ of ${\displaystyle q}$-cells in ${\displaystyle {\mathcal {A))}$ is called the ${\displaystyle q}$-th Morse number. Let ${\displaystyle \beta _{q))$ denote the ${\displaystyle q}$-th Betti number of ${\displaystyle {\mathcal {X))}$. Then, for any ${\displaystyle N>0}$, the following inequalities^[9] hold

${\displaystyle m_{N}\geq \beta _{N))$, and

${\displaystyle m_{N}-m_{N-1}+\dots \pm m_{0}\geq \beta _{N}-\beta _{N-1}+\dots \pm \beta _{0))$

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

${\displaystyle \chi ({\mathcal {X)))=m_{0}-m_{1}+\dots \pm m_{\dim {\mathcal {X))))$

Discrete Morse Homology and Homotopy Type

Let ${\displaystyle {\mathcal {X))}$ be a regular CW complex with boundary operator ${\displaystyle \partial }$ and a discrete Morse function ${\displaystyle \mu \colon {\mathcal {X))\to \mathbb {R} }$. Let ${\displaystyle {\mathcal {A))}$ be the associated Morse complex with Morse boundary operator ${\displaystyle \Delta }$. Then, there is an isomorphism^[10] of homology groups

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

and similarly for the homotopy groups.

Applications

Discrete Morse theory finds its application in molecular shape analysis,^[11] skeletonization of digital images/volumes,^[12] graph reconstruction from noisy data,^[13] denoising noisy point clouds^[14] and analysing lithic tools in archaeology.^[15]