Notes about math, research, and more.
:tags: graph-c-star-algebra, research

Outline of problem

  • k-graph (co)homology was defined by Kumjian, Pask, and Sims to yield a twisted version of k-graph C*-algebras. There is a group homomorphism from k-graph cohomology to groupoid cohomology, but it’s neither injective nor surjective in general.
  • If k-graph (co)homology were a groupoid invariant, this would give us a new way to distinguish some groups and related constructions that can be viewed as arising from k-graphs, such as Stein’s groups.
  • A place to start would be checking whether k-graph operations that are known to preserve the groupoid, such as insplitting, also preserve the k-graph (co)homology.

Papers

  1. k-graph (co)homology is defined in the two Kumjian-Pask-Sims papers
  2. The connection to groupoid (co)homology is discussed to some extent in the KPS-Twisted paper, but in more detail in the Gillaspy-Kumjian paper
  3. This is probably less relevant at the moment
  4. The Gillaspy-Wu paper might be helpful for understanding the relationship between k-graph homology and cohomology.
  5. A place to start: Is k-graph cohomology preserved by k-graph insplitting? (insplitting is defined in EFGGGP)

Things to learn about

  1. Homology
  2. Cohomology
  3. 2-cocycles
  4. “Twisted” things