Strong collapses of the Morse complex
Maxwell Lin, Nicholas A. Scoville · arXiv (Cornell University) · 2019
In this paper, we undertake an investigation of strong collapsibility and dominating vertices as they relate to the Morse complex of a simplicial complex $K$. We show that if $K$ does not contain a leaf, then its Morse complex is not strongly collapsible. If $K$ contains two leaves which share a common vertex, we are able to show that the Morse complex is strongly collapsible. We also study certain conditions under which the Morse complex strongly collapses to another Morse complex. Finally, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$, and we use this to compute the automorphism group of a disjoint union for a large collection of disjoint complexes.