the Hasse diagram {#sec: The Morse complex of the Hasse diagram}
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In this section, we will show that computing the automorphism group of ${\mathcal{M}}(K)$ is equivalent to computing the automorphism group of ${\mathcal{H}}(K)$. We will then immediately be able to compute the Morse complex of $C_n$, one of the two cases excluded in Proposition \[prop: aut(M(K))=aut(K)\]. Section \[sec: Morse complex of pd\] is then devoted to computing the Morse complex of our final special case, $K={\partial\Delta}^n.$ We briefly recall here the definition of the Hasse diagram.

The **Hasse diagram** of $K$, denoted $\mathcal{H}_K$ or $\mathcal{H}$, is defined as the partially ordered set of simplices of $K$ ordered by the face relations. We view ${\mathcal{H}}$ as a graph.

We adopt the convention that if $\sigma^{(p)}<\tau^{(p+1)}$ are two nodes of the Hasse diagram, the edge joining them is denoted $\sigma\tau$.

\[prop: aut(M(K))=aut(H(K))\] For any simplicial complex $K$, ${\mathrm{Aut}}({\mathcal{M}}(K)) \cong {\mathrm{Aut}}({\mathcal{H}}(K))$.

We will construct an isomorphism $\phi\colon {\mathrm{Aut}}({\mathcal{H}}(K)) \to {\mathrm{Aut}}({\mathcal{M}}(K))$. Consider an arbitrary automorphism $f \in {\mathrm{Aut}}({\mathcal{H}}(K)) $. Define a function\
$m \colon E({\mathcal{H}}(K)) \to V({\mathcal{M}}(K))$ by $m(\sigma\tau) = (\sigma, \tau)$, where $\sigma$ is a codimension $1$ face of $\tau$. Notice that $m$ has an inverse $m^{-1} \colon V({\mathcal{M}}(K)) \to E({\mathcal{H}}(K))$ given 