Discrete Microlocal Morse Theory

[2209.14993] Discrete Microlocal Morse Theory Skip to main content Search Submit Donate Log in Mathematics > General Topology arXiv:2209.14993 (math) [Submitted on 29 Sep 2022 ( v1 ), last revised 10 Jun 2025 (this version, v3)] Title: Discrete Microlocal Morse Theory Authors: Adam Brown , Ondrej Draganov View PDF HTML (experimental) Abstract: We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities. Comments: Added Appendix B describing the equivalence between the standard derived category and the skeleton of minimal injective chain category we define in 4.10. Changed notation of the derived functor post-composed with minimal resolution from Rf to If, to highlight the difference. Note that arXiv:2112.02609 is an old obsolete version of the paper Subjects: General Topology (math.GN) ; Computational Geometry (cs.CG); Algebraic Topology (math.AT) MSC classes: 55N30, 55N31, 55-08 Cite as: arXiv:2209.14993 [math.GN] (or arXiv:2209.14993v3 [math.GN] for this version) https://doi.org/10.48550/arXiv.2209.14993 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.jpaa.2025.108068 Focus to learn more DOI(s) linking to related resources Submission history From: Ondřej Draganov [ view email ] [v1] Thu, 29 Sep 2022 17:52:48 UTC (414 KB) [v2] Sun, 9 Jun 2024 20:54:21 UTC (1,791 KB) [v3] Tue, 10 Jun 2025 04:44:46 UTC (1,427 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GN < prev | next > new | recent | 2022-09 Change to browse by: cs cs.CG math math.AT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Discrete Microlocal Morse Theory#677300
• Selection rules for pinned and quasipinned occupation numbers with degeneracy#1050950
• Local Many Worlds in Spacetime: Deutsch-Hayden Descriptors and Local Wavefunctions#1118165
• Regularity for nonlocal problems with non-standard growth#677291
• Regularity for nonlocal problems with non-standard growth#674094