Sergei O Ivanov - On the path homology of Cayley digraphs and covering digraphs

Abstract

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a “bridge” between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph, and whose quotients have homology isomorphic to the magnitude homology.

Path homology

In pth homology sqares and triangles are important. They are directed in s special way. Path homology described the combinatorics of squares and triangles.

The box product of two digraphs is like a product but you can only move along one coordinate at atime.

Suspension: For a graph we define a suspension by adding to new vertices at the top and bottom and connect all vertices two that edge.

The fundamental grouoid ist defined as a free groupoid generated by $X$ under relations that make trianglesand squares commute.

Open questions

• Is the path cohomology algebra isomorphic to the cohomology algebra of a space?
• Is the path cohomology algebra graded commutative?
• Is the path cohomology of a box product and strong box product isomorphic

The author restricted himself to Cayley-Graphs and found a connection between path homology and group homology.

l-covering digraphs

We want to defined a $l$-covering digraph, s.t. the subgroups of ${\pi }_1^l(X)$ and the $l$-covering digraphs of $X$ are isomorphic.

A morphisms od digraph is an $l$-covering when special conditions are fulfilled. It feels like this is a generalization of topological coverings.

There is a definition in terms of metric fibrations. (Ask the author for mor information.)

Cayley Digraphs

Consider the additive group of rational numbers generated by the inverse of factorials.