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 Aut(M(K))≅Aut(K). In the case where K=Cn, a cycle of length n, we show that Aut(M(Cn))≅Aut(C2n). When K=∂Δn, we prove that Aut(M(∂Δn))≅Aut(∂Δn)×Z2. These results are based on recent work of Capitelli and Minian.