Higher connectivity of the Morse complex
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- by Nicholas A. Scoville and Matthew C. B. Zaremsky HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 9 (2022), 135-149
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.”References
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Additional Information
- Nicholas A. Scoville
- Affiliation: Department of Mathematics and Computer Science, Ursinus College, Collegeville, Pennsylvania 19426
- MR Author ID: 792053
- ORCID: 0000-0002-4736-3605
- Email: nscoville@ursinus.edu
- Matthew C. B. Zaremsky
- Affiliation: Department of Mathematics and Statistics, University at Albany (SUNY), Albany, New York 12222
- MR Author ID: 852515
- Email: mzaremsky@albany.edu
- Received by editor(s): April 27, 2020
- Received by editor(s) in revised form: July 24, 2020, August 18, 2020, and November 17, 2021
- Published electronically: April 12, 2022
- Additional Notes: The second author was supported by grant #635763 from the Simons Foundation.
- Communicated by: Patricia Hersh
- © Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 9 (2022), 135-149
- MSC (2020): Primary 55U05; Secondary 57Q05
- DOI: https://doi.org/10.1090/bproc/115
- MathSciNet review: 4407041