Discrete Microlocal Morse Theory
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.
Keywords
Cite
@article{arxiv.2209.14993,
title = {Discrete Microlocal Morse Theory},
author = {Adam Brown and Ondrej Draganov},
journal= {arXiv preprint arXiv:2209.14993},
year = {2025}
}
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