Bigraded path homology and the magnitude-path spectral sequence
演讲者
Emily Roff
时间
2024年04月03日 14:30 至 15:30
地点
A3-4-101
线上
Zoom 559 700 6085
(BIMSA)
摘要
The past decade has seen large literatures develop around two novel invariants of directed graphs: magnitude homology (due to Leinster, Hepworth and Willerton) and the path homology of GLMY theory. Though their origins are quite separate, Asao proved in 2022 that in fact these homology theories are intimately related. To every directed graph one can associate a certain spectral sequence - the magnitude-path spectral sequence, or MPSS - whose page $E^1$ is exactly magnitude homology, while path homology lies along a single axis of page $E^2$.
This talk has two subjects: the MPSS as a whole, and its page $E^2$, which we call the bigraded path homology of a directed graph. I will explain the construction of the sequence and argue that each one of its pages deserves to be regarded as a homology theory for digraphs, satisfying a Künneth formula and an excision theorem, and with a homotopy-invariance property that grows stronger as we turn the pages of the sequence. The second half of the talk will focus on bigraded path homology, which shares the important homological properties of ordinary path homology but is a strictly finer invariant - capable of distinguishing, for example, the directed n-cycles for all $n > 2$. We will close with some speculations on the implications of all this for the formal homotopy theory of the category of directed graphs.
This talk has two subjects: the MPSS as a whole, and its page $E^2$, which we call the bigraded path homology of a directed graph. I will explain the construction of the sequence and argue that each one of its pages deserves to be regarded as a homology theory for digraphs, satisfying a Künneth formula and an excision theorem, and with a homotopy-invariance property that grows stronger as we turn the pages of the sequence. The second half of the talk will focus on bigraded path homology, which shares the important homological properties of ordinary path homology but is a strictly finer invariant - capable of distinguishing, for example, the directed n-cycles for all $n > 2$. We will close with some speculations on the implications of all this for the formal homotopy theory of the category of directed graphs.