Pure Math Graduate Student Seminar: Kylan Schatz, NC State, Abelian 3-cocycles from Quadratic Forms – Quinn’s Formula

Name: Pure Math Graduate Student Seminar: Kylan Schatz, NC State, Abelian 3-cocycles from Quadratic Forms – Quinn’s Formula
Start: 2022-10-17T15:00:00-04:00
End: 2022-10-17T16:00:00-04:00
Location: SAS 2235

October 17, 2022 | 3:00 pm - 4:00 pm EDT

For some nice subclass of braided monoidal categories, the associator and braiding data are described by an Abelian 3-cocycle $[(h,c)]$ . Eilenberg and MacLane demonstrated a surprisingly simple isomorphism between the groups of Abelian 3-cocycles with coefficients in $A$ and quadratic forms with coefficients in $A$ , the trace map $\mathrm{tr}:H^3_{ab}(G, A) \to \mathrm{Quad}(G, A)$ by $\mathrm{tr}[(h,c)](x) = c(x,x)$ . Like many category theoretical results, this was proven at the level of topology. In this talk, I will elaborate how to algebraically reconstruct an Abelian 3-cocycle $(h,c)$ given a quadratic form $q$ following results from arXiv:2005.05243.