On the automorphism group of the Morse complex

Let $K$ be a finite, connected, abstract simplicial complex. The Morse complex of $K$, first introduced by Chari and Joswig, is the simplicial complex constructed from all gradient vector fields on $K$. We show that if $K$ is neither the boundary of the $n$-simplex nor a cycle, then $\mathrm{Aut}(\mathcal{M}(K))\cong \mathrm{Aut}(K)$. In the case where $K= C_n$, a cycle of length $n$, we show that $\mathrm{Aut}(\mathcal{M}(C_n))\cong \mathrm{Aut}(C_{2n})$. In the case where $K=\partial\Delta^n$, we prove that $\mathrm{Aut}(\mathcal{M}(\partial\Delta^n))\cong \mathrm{Aut}(\partial\Delta^n)\times \mathbb{Z}_2$. These results are based on recent work of Capitelli and Minian.