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On The Evans Chain Complex

S. Joseph Lippert

TL;DR

This work advances the computation of $K$-theory for higher-rank graph $C^*$-algebras by refining Evans's chain complex into a tractable block-matrix form. It reveals a recursive, tensor-product structure $\mathcal{D} \cong \bigotimes_{j=1}^k \mathcal{C}^j$ for the Evans complex and leverages the K"unneth theorem to obtain explicit homology and $K$-theory results, including a vanishing criterion when $\gcd(B_1,\dots,B_k)=1$. In the one-vertex case, the approach yields concrete $K$-theory computations and highlights how invertibility properties of the co-adjacency matrices govern the $K_*$ groups. Overall, the block-matrix framework provides practical tools for identifying trivial $K$-theory and performing explicit homology calculations via spectral sequences.

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 Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

On The Evans Chain Complex

TL;DR

This work advances the computation of -theory for higher-rank graph -algebras by refining Evans's chain complex into a tractable block-matrix form. It reveals a recursive, tensor-product structure for the Evans complex and leverages the K"unneth theorem to obtain explicit homology and -theory results, including a vanishing criterion when . In the one-vertex case, the approach yields concrete -theory computations and highlights how invertibility properties of the co-adjacency matrices govern the groups. Overall, the block-matrix framework provides practical tools for identifying trivial -theory and performing explicit homology calculations via spectral sequences.

Abstract

We elaborate on the construction of the Evans chain complex for higher-rank graph -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 -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 Künneth theorem to explicitly compute the homology groups of the Evans chain complex.
Paper Structure (4 sections, 12 theorems, 33 equations, 1 figure)

This paper contains 4 sections, 12 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Block Matrix Presentation
  4. Appendix

Key Result

Theorem 2.6

evans08The Evans Chain Complex: For $p,k\in \mathbb{N}$ with $0\leq p \leq k$ define the set Equip these sets with the standard map such that for $a\in \mathbb{N}_{p,k}$ the element $a^i\in \mathbb{N}_{p-1,k}$ is the tuple $a$ with the $i^{th}$ coordinate removed. Additionally, $N_{0,k}:=\{*\}$ with $a^1=*$ for all $a\in N_{1,k}$. For a row-finite, source-free $k$-graph, $\Lambda$, there ex where

Figures (1)

  • Figure 3.4: The differential depicted are $\partial_3^4$ and $\partial_4^4$ respectively. Each position is made zero if there is no way to delete an element of the column index and obtain the row index ($\delta_{b,a^i}$). If an even position needs to be deleted, a negative is added ($(-1)^{i+1}$). Finally, whatever element was deleted determines which $B_i$ is put in the slot ($B_{a_i}$).

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 18 more