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Basic information

Accomplishments based on abstract

  1. Introduce homology theory for $K$-graphs and explore its fundamental properties.
  2. Establish connections with algebraic topology
    • Show that the homology of a $K$-graph coincides with the homology of its opological realization (as described by Kaliszewski et al.)
  3. Show how to twist the \(C^\ast\text{-algebra}\) of a $K$-graph by a \(\mathbb{T}\text{-valued}\) 2-cocyle and demonstrat e that examples include all noncommutative tori.

Introduction

  • Motivation homology theory promices to have an interesting application to \(C^\ast\text{-algebras.}\) Application discussed in Section 7.
  • In this paper they connect the study of homology of $K$-graphs.
  • Section 2:
    • basic definitions

Section 3:

  • define homology
  • prove it is a functor
  • show that you can measure connectedness by the 0th homology group
  • show that the 1-cycles correspond naturally to integer combinations of undirected cycles in the $K$-graph.

Section 4:

  • Prove analogues of a number of standard theorems in algebraic topology for their homology
  • Show that every automorphism of a $K$-graph induces a long exact sequence in homology which corresponds exactly to the long exact sequence for a mapping torus.

Section 5:

  • Describe examples of 2-graphs whose homology is identical to that of the sphere, the torus, the Klein bottle and the projective plane respectively.

Section 6:

  • Show that the homology for a $K$-graph agrees with the singular homolgy of its topological realization. This suggests strongly that our homology theory is a reasonable one for $K$-graphs.

Section 7:

  • Discuss twisted $K$-graph \(C^\ast\text{-algebras.}\)
    • Was motivation for the homology of $K$-graphs
  • Introduce the notion of the \(C^\ast\text{-algebra}\) of $K$-graph twisted by a \(\mathbb{T}\text{-valued}\) 2-cocycle, and show that the isomorphism class of the \(C^\ast\text{-algebra}\) depends only on the cohomology class of the cocycle.
  • Some basic examples of finite $K$-graphs whose twisted \(C^\ast\text{-algebras}\) capture the noncummutative tori and the Heegard-type quantum 3-spheres of some citation.

Notation

  • \(\Lambda\) is $k$-graph.
  • If \(\lambda \in \Lambda\) then \(\lambda\) is a vertex.
  • If \(n \in \mathbb{N}^k\) then \(|n| = \sum_{i=1}^k n_{i}\).
  • \(\Lambda^0\) is the set of vertices?
  • \(\Lambda^1\) is the set of edges?
  • \(\mu, \nu\) are paths typically. A path is a list of edges. Then \(\mu_{i}\) would be a single edge.