Abstract
Important recent work of Asao conclusively demonstrated that magnitude homology and path homology of graphs, both previously unrelated, are in fact just two aspects of a much larger object, the magnitude-path spectral sequence or MPSS. Magnitude homology is precisely the E1 page of this sequence, while path homology is an axis of the E2 page. In this talk I will present joint work in progress with Emily Roff. We show that the E2 page of the MPSS satisfies Kunneth, excision and Mayer-Vietoris theorems. These, together with the homotopy-invariance property proved by Asao, show that the entire E2 term should be regarded as a homology theory, which we call “bigraded path homology”. Second, we show that bigraded path homology is a strictly stronger invariant than path homology, by demonstrating that it can completely distinguish the directed cycles, none of which can be told apart under the original path homology.
Talk
We study Reachability chains. Compare this to Erreichbarkeitshomologie.