# On the automorphism group of the Morse complex

## TL;DR

This paper investigates the automorphism group of the Morse complex derived from a finite simplicial complex, establishing isomorphisms with the automorphism group of the original complex for most cases, and identifying special cases with additional symmetries.

## Contribution

It characterizes the automorphism group of the Morse complex for various classes of simplicial complexes, revealing when it coincides with or extends the automorphism group of the original complex.

## Key findings

- For most complexes, Aut(M(K)) is isomorphic to Aut(K).
- For cycles, Aut(M(C_n)) is isomorphic to Aut(C_{2n}).
- For boundary complexes of simplices, Aut(M(∂Δ^n)) is isomorphic to Aut(∂Δ^n)×Z_2.

## 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.