Bridging between überhomology and double homology
Überhomology is a recently defined triply-graded homology theory of simplicial complexes, which yields both topological and combinatorial information. When restricted to (simple) graphs, a certain specialization of überhomology gives a categorification of the connected domination polynomial at -1; which shows that it is related to combinatorial quantities. On the topological side, überhomology detects the fundamental class of homology manifolds, showing that this invariant is a mixture of both. We will show that (a specialization of) überhomology of simplicial complexes can be identified with the second page of the Mayer-Vietoris spectral sequence, with respect to the anti-star covers. As a consequence, this provides a connection between überhomology and the double homology of moment angle complexes, as defined by Limonchenko-Panov-Song-Stanley. This is joint work with D. Celoria and C. Collari.