Double, über and poset homology
Carlos Gabriel Valenzuela Ruiz
TL;DR
The paper compares two homology theories associated to a simplicial complex $\mathcal{K}$: uberhomology and the double homology of its moment-angle complex. It constructs and analyzes maps $\phi_{l,q}$ between the corresponding poset-cohomology expressions, proving isomorphisms for $l>2$ or $q>0$ and deriving an exact sequence in degree $(2,0)$. It further shows that the complete difference between the theories is captured by a simple, neighbourliness-dependent formula for the bigraded Poincaré series. Overall, the work provides a precise, case-by-case correspondence between uberhomology and double homology and highlights a sharp dichotomy dictated by neighbourliness.
Abstract
We present a comparison map between the uberhomology of a simplicial complex $\mathcal{K}$ and the double homology of its associated moment-angle complex $\mathcal{Z}_{\mathcal{K}}$. We show these two homology theories differ at three bidegrees, which depend on whether the complex $K$ is neighbourly or not.