\sigma  =0$, so $f(\sigma) \in V(B)$. Likewise, we know $\dim f(\sigma) = \dim g(f(\sigma))$, so $g(f(\sigma)) \in V(C)$. Therefore, $(g\circ f)_V(\sigma) = (g\circ f)(\sigma) = (g \circ f_V)(\sigma) = (g_V\circ f_V)(\sigma),$ as desired.

A simplicial map which is a bijection is a **simplicial isomorphism**, and if $f\colon K \to K$ is a simplicial isomorphism, we say that $f$ is a **simplicial automorphism**. The **automorphism group** of $K$ is defined by $${\mathrm{Aut}}(K):=\{f\colon K \to K \mid f \text{ is an automorphism}\}.$$

Because we need to refer to them below, we define a cycle $C_n$ and the boundary of the $n$-simplex ${\partial\Delta}^n.$

Let $n \geq 3$ be an integer. Define the **cycle of length $n$**, denoted $C_n$, to be the $1$-dimensional simplicial complex (graph) with vertex set $V(C_n):=\{v_0, v_1, \ldots, v_{n-1}\}$ and edge set given by $$\{v_0,v_1\}, \{v_1,v_2\}, \{v_2, v_3\}\ldots, \{v_{n-2}, v_{n-1}\}, \{v_{n-1},v_0\}.$$

Let $n\geq 1$ be an integer. The **boundary of the $n$-simplex**, denotes ${\partial\Delta}^n$ is the simplicial complex given by ${\partial\Delta}^n:=\mathcal{P}([v_n])- \{\emptyset, \{v_0,\ldots, v_{n}\}\}.$

The Morse complex
-----------------

In this section, we recall the basics of the Morse complex. Our references for discrete Morse theory in general are [@F-02; @KnudsonBook; @KozlovBook] and the Morse complex in particular are [@CJ-2005; @CM-17].

