On The Evans Chain Complex

TL;DR

This paper develops a new block matrix approach to the Evans chain complex for higher-rank graph $C^*$-algebras, enabling the identification of algebras with trivial K-theory and explicit homology computations in special cases.

Contribution

Introduces a block matrix presentation of the Evans chain complex differential maps for higher-rank graph $C^*$-algebras, facilitating new K-theory and homology results.

Findings

Identifies a broad class of higher-rank graph $C^*$-algebras with trivial K-theory.

Uses the Künneth theorem to explicitly compute homology groups for one-vertex higher-rank graphs.

Provides a new computational framework for the Evans chain complex.

Abstract

We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialized case where the higher-rank graph consists of one vertex, we are able to use the K\"unneth theorem to explicitly compute the homology groups of the Evans chain complex.