C_n$ or ${\partial\Delta}^n$. The next two sections compute ${\mathrm{Aut}}({\mathcal{M}}(K))$ in the two excluded cases.

Induced maps on ${\mathcal{M}}(K)$ {#Induced Maps}
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If $f_V\colon V(K) \to V(L)$ is a bijection on the vertex sets and induces a simplicial map $f\colon K\to L$, then $f$ is an isomorphism. We are interested in isomorphisms from $K$ to $K$, i.e., automorphisms of $K$. Given an automorphism of $K$, we now define an induced automorphism on ${\mathcal{M}}(K)$.

Let $f\colon K\to K$ be an automorphism. Define the **induced automorphism on the Morse complex** $f_{\ast_V}\colon V(\mathcal{M}(K)) \to V(\mathcal{M}(K))$ by $f_{\ast_V}(v) = (f(\sigma), f(\tau))$ where $v = (\sigma^{(p)}, \tau^{(p+1)}) \in V(\mathcal{M}(K))$.

We then extend $f_{\ast_V}$ to a simplicial map on all of ${\mathcal{M}}(K)$. Below we will justify our claims that this yields a well-defined automorphism of ${\mathcal{M}}(K)$.

\[prop: primitive to primitive\] If $v =(\sigma^{(p)}, \tau^{(p+1)})  \in V(\mathcal{M}(K))$, then $f_{\ast_V}(v) \in V(\mathcal{M}(K))$.

We seek to show that $(f(\sigma), f(\tau))$ is a primitive vector of $K$. Since $f_V$ is a bijection, we know that $\dim f(\sigma) = \dim \sigma$ and $\dim f(\tau) = \dim \tau$. Hence we have $\dim f(\sigma) = \dim\sigma = \dim\tau-1 =\dim f(\tau) - 1$. Since $\sigma \subseteq \tau$, $f(\sigma)\subseteq f(\tau)$ so that $(f(\sigma), f(\tau))$ is primitive 