Bridging between \"uberhomology and double homology

TL;DR

This paper proves an isomorphism between "uberhomology and double homology for finite simplicial complexes, clarifying their relationship and demonstrating how "uberhomology detects simplexes and relates to the connected domination polynomial.

Contribution

It establishes a formal isomorphism between "uberhomology and double homology, linking these theories through spectral sequences and providing new insights into their properties.

Findings

"Uberhomology detects the standard simplex.

Double homology's diagonal relates to the connected domination polynomial.

Theories are connected via a Mayer-Vietoris spectral sequence.

Abstract

We establish an isomorphism between the 0-degree \"uberhomology and the double homology of finite simplicial complexes, using a Mayer-Vietoris spectral sequence argument. We clarify the correspondence between these theories by providing examples and some consequences; in particular, we show that \"uberhomology groups detect the standard simplex, and that the double homology's diagonal is related to the connected domination polynomial.