The geometric realization of regular path complexes via (co-)homology

TL;DR

This paper establishes a geometric realization of regular path complexes through (co)homology, linking them to singular complexes and Hochschild (co)homology, and extends classical algebraic topology results to these structures.

Contribution

It introduces a geometric realization of regular path complexes via singular $ riangle$-complexes, connecting path (co)homology to Hochschild (co)homology and extending Eilenberg-Zilber and Künneth theorems.

Findings

Path (co)homology is isomorphic to that of associated singular complexes.

Path (co)homology coincides with Hochschild (co)homology for regular finite complexes.

Eilenberg-Zilber and Künneth formulas are valid for regular path complexes.

Abstract

The aim of this paper is to give the geometric realization of regular path complexes via (co)homology groups with coefficients in a ring $R$. Concretely, for each regular path complex $P$, we associate it with a singular $\Delta$-complex $S(P)$ and show that the (co)homology groups of $P$ are isomorphic to those of $S(P)$ with coefficients in $R$. As a direct result we recognize path (co)homology as Hochschild (co)homology in case that $R$ is commutative and $P$ regular finite. Analogues of the Eilenberg-Zilber theorem and K\"{u}nneth formula are also showed for the Cartesian product and the join of two regular path complexes. In fact, we meanwhile improve some previous results which are covered by these conclusions in this paper.