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arXiv · 2004.10481

Higher connectivity of the Morse complex

Nicholas A. Scoville·Matthew C. B. Zaremsky
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Abstract

The Morse complex $\mathcal{M}(\Delta)$ of a finite simplicial complex $\Delta$ is the complex of all gradient vector fields on $\Delta$. In this paper we study higher connectivity properties of $\mathcal{M}(\Delta)$. For example, we prove that $\mathcal{M}(\Delta)$ gets arbitrarily highly connected as the maximum degree of a vertex of $\Delta$ goes to $\infty$, and for $\Delta$ a graph additionally as the number of edges goes to $\infty$. We also classify precisely when $\mathcal{M}(\Delta)$ is connected or simply connected. Our main tool is Bestvina-Brady Morse theory, applied to a "generalized Morse complex."

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Nicholas A. Scoville, Matthew C. B. Zaremsky. 2020-04-22. Higher connectivity of the Morse complex. https://arxiv.org/abs/2004.10481

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