Discrete Microlocal Morse Theory

TL;DR

This paper develops a computational framework combining discrete Morse theory and microlocal sheaf theory for finite posets and simplicial complexes, introducing algorithms and bounds for sheaf resolutions and a new discrete microsupport concept.

Contribution

It introduces a novel discrete microsupport, algorithms for minimal injective resolutions, and extends Morse theory to a microlocal setting in finite posets.

Findings

Unique minimal injective resolutions for sheaves on finite posets

Algorithms for computing sheaf resolutions and functors

Asymptotically tight bounds on resolution complexity

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.