## Scattered Thoughts

A deep result due to Dijkgraaf and Witten states that the Chern-Simons actions for a compact gauge group G are in one–to–one correspondence with the elements of the cohomology group $H^4(BG, Z)$ of the classifying space BG with integer coeﬃcients Z. In particular, this classiﬁcation includes the case of ﬁnite groups H. The isomorphism $H^4(BH, Z)\simeq H^3(H, U(1))$ which is only valid for ﬁnite H then implies that the diﬀerent CS theories for a ﬁnite gauge group H correspond to the diﬀerent
elements $\omega \in H^3(H, U(1))$, i.e. algebraic 3-cocycles ω taking values in U(1).