From EG: I think probably the best way to start is, yes, with k-graph homology, and with an example in mind. As I said in an earlier email, I think the first question is: Does k-graph insplitting preserve k-graph homology? So, I would start by picking one of the examples of insplitting from EFGGGP, and compute the k-graph homology (as defined in KumjianPaskSimsHomology, Section 7, or equivalently in Gillaspy-Wu Section 3).

1. Understand \(k\)-graphs
2. Understand \(k\)-graphs homology (get a feel for it)
   • Defined in Kumjian Pask Sims “Homology…” paper
3. Understand insplitting
   1. pick example from EFGGGP
      1. They have graphs and their insplitting
   2. compute the \(k\)-graph homology
4. QUESTION: Does insplitting preserve \(k\)-graph homology?
5. Has EG calculated much \(k\)-graph homology?

EG thought about this a year ago and did a couple of examples

1. EFGGP - moves on \(k\)-graphs preserving morita equivalence
2. Gillaspy Wu Kumjian - cohomology for small categories
3. Gillaspy and Wu - Cubical categorical cohomology
4. Kumjian Pask Sims - Homology for higher-rank graphs and twisted \(C^*\)-algebras
5. Kumjian Pask Sims - On twisted higher-rank graph \(C^*\)-algebras

1. rp k graphs
2. rp k-graph master file