then generates all the ghost automorphisms of the Morse complex of ${\partial\Delta}^n$. By studying these ghost automorphisms, we account for all automorphisms of ${\mathcal{M}}({\partial\Delta}^n)$.

The outline of this paper is as follows: Section \[sec: background\] gives necessary background, terminology, and notation. Section \[sec: The Automorphism group\] is the heart of the paper where we compute the automorphism group of ${\mathcal{M}}(K)$. In Section \[Induced Maps\], we show that any automorphism $K$ induces an automorphisms of ${\mathcal{M}}(K)$ so that there is an injective homomorphism ${\mathrm{Aut}}(K) \to {\mathrm{Aut}}({\mathcal{M}}(K))$. This injection turns out to be an isomorphism in the case where in the case where $K\neq C_n$ or ${\partial\Delta}^n$ by Proposition \[prop: aut(M(K))=aut(K)\]. We then turn to the cases $K=C_n$ and $K={\partial\Delta}^n$. We show that ${\mathrm{Aut}}({\mathcal{M}}(C_n))\cong {\mathrm{Aut}}(C_{2n})$ in Section \[sec: The Morse complex of the Hasse diagram\]. This follows fairly easily from the fact that ${\mathrm{Aut}}({\mathcal{M}}(K))\cong {\mathrm{Aut}}({\mathcal{H}}(K))$ (Proposition \[prop: aut(M(K))=aut(H(K))\]). Finally, Section \[sec: Morse complex of pd\] is devoted to computing the automorphism group of ${\partial\Delta}^n$ via the ghost automorphisms mentioned above.

Background {#sec: background}
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All our simplicial complexes are assumed to be abstract, finite, and connected simplicial complexes. Our reference for the basics of simplicial complexes is [@F-P11] or [@J-11].

Let $n\geq 