Abstract:

We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories, closely related to discrete Morse matchings on X.  We show that, by placing these homologies in a poset, we obtain a triply graded homology theory which we call überhomology. This latter homology is not a homotopy invariant, but nonetheless encodes both combinatorial and topological information on X. Time permitting we'll talk about a recent collaboration with Caputi and Collari, relating a specialisation of the überhomology to connected dominating sets in graphs.