A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes
Abstract
We investigate the homology of cosheaves over finite simplicial complexes. After constructing the Mayer-Vietoris short exact sequence for this homology theory, we apply discrete Morse theory to this setting, defining the associated Morse chain complex. The main result is the construction of a Morse Mayer-Vietoris short exact sequence of Morse chain complexes, for which we establish a quasi-isomorphism to the standard Mayer-Vietoris sequence. Discussions relating poset-based (co)sheaf definitions to topological ones via Kan extensions and exploring a modern categorical perspective on discrete Morse theory are also included.
Keywords
Cite
@article{arxiv.2508.14103,
title = {A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes},
author = {Ben H. Gould},
journal= {arXiv preprint arXiv:2508.14103},
year = {2025}
}
Comments
Essay submitted for the C3.9 Computational Algebraic Topology course (2025), towards a Masters of Mathematics (MMath) at the University of Oxford