# Discrete Morse Theory

### Simplex

A simplex is just an $$n$$-dimensional “triangle” formally

$\Delta^n=\LS x \in \R^k: x_0+\cdots+x_{k-1}=1, x_i \geq 0 \text { for } i=0, \ldots, k-1\RS$

We can also describe an $$n$$ simplex combinatorially as a list of the vertices of the simplex $\sigma = \LB e_0,\ e_1,\ \cdots, e_n\RB$

### Face of a Simplex

We define a face $$\tau$$, of a simplex $$\sigma$$, as a proper subset of $$\sigma$$ that is

$\tau < \sigma$ when $\LB a,b\RB \subset \LB a, \ b,\ c\RB$

### Simplicial complex

A simplicial complex is a space $$K$$ built from glued together simplices and must satisfy

1. If $$\tau$$ is the face of a simplex $\sigma \in K$ then $$\tau\in K$$
2. If $$\sigma, \tau \in K$$ then $$\sigma\cap \tau$$ is a face of $$\sigma$$ and $$\tau$$.  It’s convenient to have a combinatorial description for $$K$$. This description is called an “abstract simplicial complex” and is just the collection of combinatorial descriptions of simplices of $$K$$ $\LS \LB b,c,d\RB, \LB b,c\RB, \LB b,d\RB, \LB c,d\RB, \LB a,c\RB, \LB a,b\RB, \LB d,e\RB, \LB a\RB, \LB b\RB, \LB c\RB, \LB d\RB, \LB e\RB\RS$

### Definition of a “Discrete function”

A discrete function $$f$$ on a simplicial complex $$K$$ assigns a number to each simplex

\begin{aligned} f\LP\LB b,c,d\RB\RP &= 5 \\ f\LP\LB b,c\RB\RP &= 4 \\ f\LP\LB b,d\RB\RP &= 5 \\ f\LP\LB c,d\RB\RP &= 3 \\ f\LP\LB a,c\RB\RP &= 1 \\ f\LP\LB a,b\RB\RP &= 1 \\ f\LP\LB d,e\RB\RP &= 5 \\ f\LP\LB a\RB\RP &= 0 \\ f\LP\LB b\RB\RP &= 2 \\ f\LP\LB c\RB\RP &= 2 \\ f\LP\LB d\RB\RP &= 4 \\ f\LP\LB e\RB\RP &= 6 \\ \end{aligned} ### Definition of a “Discrete Morse function”

A discrete function $f:K\to \R$ is called a discrete Morse function if for every $$n$$-simplex $$\alpha^n \in K$$ the following are true

1. $$\#\LS \alpha^n<\beta^{n+1} \MM f\LP\beta^{n+1}\RP\leq f\LP\alpha^n\RP\RS\leq 1$$
2. $$\#\LS \alpha^n>\gamma^{n-1} \MM f\LP\gamma^{n-1}\RP\geq f\LP\alpha^n\RP\RS\leq 1$$
1. $$\#\LS \alpha^n<\beta^{n+1} \MM f\LP\beta^{n+1}\RP\leq f\LP\alpha^n\RP\RS\leq 1$$
2. $$\#\LS \alpha^n>\gamma^{n-1} \MM f\LP\gamma^{n-1}\RP\geq f\LP\alpha^n\RP\RS\leq 1$$ 1. $$\#\LS \alpha^n<\beta^{n+1} \MM f\LP\beta^{n+1}\RP\leq f\LP\alpha^n\RP\RS\leq 1$$
2. $$\#\LS \alpha^n>\gamma^{n-1} \MM f\LP\gamma^{n-1}\RP\geq f\LP\alpha^n\RP\RS\leq 1$$ ### Definition of a Critical simplex

An $$n-$$simplex $$\alpha^n$$, of a complex $$K$$, is critical under a discrete Morse function $$f$$ if the following are true

1. $$\#\LS \alpha^n<\beta^{n+1} \MM f\LP\beta^{n+1}\RP\leq f\LP\alpha^n\RP\RS=0$$
2. $$\#\LS \alpha^n>\gamma^{n-1} \MM f\LP\gamma^{n-1}\RP\geq f\LP\alpha^n\RP\RS=0$$

We should note that for all the $$\alpha\in K$$ at least one of $$1.$$ and $$2.$$ must be true.

1. $$\#\LS \alpha^n<\beta^{n+1} \MM f\LP\beta^{n+1}\RP\leq f\LP\alpha^n\RP\RS=0$$
2. $$\#\LS \alpha^n>\gamma^{n-1} \MM f\LP\gamma^{n-1}\RP\geq f\LP\alpha^n\RP\RS=0$$ A discrete gradient vector field on a simplicial complex $$K$$ is a collection of pairs $$V=\LS\tau^n<\sigma^{n+1}\RS$$ of simplices in $$K$$ such that each simplex is in at most one pair of $$V$$.

A discrete Morse function induces a gradient vector field on $$K$$.

If $$\alpha$$ is a non-critical simplex, and $\alpha<\beta$ with $f\LP\beta\RP<f\LP\alpha\RP$ We draw an arrow from $$\alpha\to \beta$$. ### Simplicial collapse

The simplicial equivalent of a deformation retraction is called a simplicial collapse.

### Maximal Face

A face $$\sigma\in K$$ is called maximal if there is no $$\beta\in K$$ so that $$\sigma<\beta$$. ### Free face

A simplex $$\tau$$ of a simplicial complex $$K$$ is called a free face if $$\tau$$ is a face of exactly one maximal face. ### Collapse

If $$\tau$$ is a free face of $$\sigma$$ we can collapse $$K \searrow K^\prime$$ by the following set operation

$K^\prime = K-\LP\tau \cup \sigma\RP$ ### Collapsing A Vector Field

Given a vector field we can simplicially collapse following the arrows. ### Main result of discrete Morse Theory

Suppose $$K$$ is a simplicial complex with a discrete Morse function. Then $$K$$ is homotopy equivalent to a CW complex with exactly one cell of dimension $$n$$ for each critical simplex of dimension $$n$$.

## Star of a face

The star of a face $$\tau$$ of $$K$$ is the subcomplex of $$K$$ consisting of all faces $$\sigma$$ of $$K$$ where $$\tau < \sigma$$, as well as all faces of $$\sigma$$.  The link of a face $$\tau$$ of $$K$$ is the subcomplex of $$K$$ consisting of all faces of the star of $$\tau$$ that do not intersect $$\tau$$.  ### Combinatorial d-ball

A complex $$K$$ is a combinatorial d-ball if $$K$$ and $$\Delta^d$$ have isomorphic subdivisions.

A combinatorial $$\LP d-1\RP$$-sphere is the boundary of a $$d$$-ball

### combinatorial d-manifold

A complex is a $$d$$-manifold if the link of every vertex is either a $$(d-1)$$-ball or $$(d-1)$$-sphere.

### Sphere theorems

Let $$K$$ be a combinitorial $$d$$-manifold with boundary which simplicially collapses to a vertex. Then $$K$$ is a combinitorial $$d$$-ball.

Let $$X$$ be a combinatorial $$d$$-manifold with a discrete with a discrete Morse function with exactly two critical simplices. Then $$X$$ is a combinitorial sphere.