Mathematics > General Topology
[Submitted on 29 Sep 2022 (v1), last revised 9 Jun 2024 (this version, v2)]
Title:Discrete Microlocal Morse Theory
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.
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)
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