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Basic information
- By Alex Kumjian, David Pask, and Aidan Sims.
- Introduced homology for k-graphs and twisted \(C^\ast\text{}-algebras\)}.
Accomplishments based on abstract
- Introduce homology theory for $K$-graphs and explore its fundamental properties.
- 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.)
- 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.