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abstract: '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.'
author:
- 'Maxwell Lin, Nicholas A. Scoville'
date: April 2019
title: On the automorphism group of the Morse complex
---

*Keywords:* Discrete Morse theory, automorphism group, Morse complex, gradient vector field\
*2000 MSC:*Primary: 55U05, 08A35 Secondary: 52B05, 57Q05

Introduction
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In 2005, Chari and Joswig [@CJ-2005] introduced the Morse complex of a simplicial complex. The Morse complex is based on Forman’s discrete Morse theory [@F-95; @F-02] where after fixing a simplicial complex $K$, one builds a new simplicial complex ${\mathcal{M}}(K)$ from the collection of all gradient vector fields or arrows on $K$. Chari and Joswig computed the homotopy type of the Morse complex when $K$ is the $n$-simplex. Ayala et 