Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bridging between überhomology and double homology

Luigi Caputi, Daniele Celoria, Carlo Collari

TL;DR

The paper proves an isomorphism between the $0$-degree überhomology of a finite, connected simplicial complex and the double homology of its moment-angle complex, using the Mayer-Vietoris spectral sequence with respect to the anti-star cover. The main result identifies the two theories (via a degree-preserving correspondence) as the second page of a Mayer-Vietoris construction, enabling a transfer of computational and structural insights between combinatorial and topological invariants. Consequences include that überhomology detects simplices and that the diagonal of double homology relates to graph-theoretic invariants such as the connected domination polynomial. The work also clarifies limitations (notably, the $i=0$ case) and provides concrete examples to illustrate the correspondence. Overall, it offers a unified framework linking combinatorial and moment-angle complex homologies and suggests further directions for interpreting full überhomology through MV-theoretic data.

Abstract

We establish an isomorphism between the 0-degree überhomology 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 überhomology groups detect the standard simplex, and that the double homology's diagonal is related to the connected domination polynomial.

Bridging between überhomology and double homology

TL;DR

The paper proves an isomorphism between the -degree überhomology of a finite, connected simplicial complex and the double homology of its moment-angle complex, using the Mayer-Vietoris spectral sequence with respect to the anti-star cover. The main result identifies the two theories (via a degree-preserving correspondence) as the second page of a Mayer-Vietoris construction, enabling a transfer of computational and structural insights between combinatorial and topological invariants. Consequences include that überhomology detects simplices and that the diagonal of double homology relates to graph-theoretic invariants such as the connected domination polynomial. The work also clarifies limitations (notably, the case) and provides concrete examples to illustrate the correspondence. Overall, it offers a unified framework linking combinatorial and moment-angle complex homologies and suggests further directions for interpreting full überhomology through MV-theoretic data.

Abstract

We establish an isomorphism between the 0-degree überhomology 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 überhomology groups detect the standard simplex, and that the double homology's diagonal is related to the connected domination polynomial.

Paper Structure

This paper contains 1 section, 2 theorems, 10 equations, 1 figure.

Table of Contents

  1. $0$-degree überhomology

Key Result

Theorem 1

Let $K$ be a finite and connected simplicial complex with $m$ vertices. Then, there is an isomorphism for each $i\neq 0,-1$. Equivalently, for each choice of $l$ and $k$ such that $l-k \neq 0, 1$.

Figures (1)

  • Figure 1: For a simplicial complex $K$ with $3$ vertices, the horizontal homologies of $K$ are placed over the vertices of a 3d cube, whose edges are components of the überhomology differentials.

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Example 1.5