# Dijkgraaf-Witten topological invariant

We know that given a finite group $G$ and its group 4-cohomology class $w \in H^4[G;U(1)]$, we can obtain a DW topological invariant $Z_{G,w}(M^4)$ as the partition function of the DW theory on a closed 4-manifold $M^4$.

Is it possible that a finite group $G$ may have two different 4-cohomology classes $w$ and $u$, such that they give rise to the same DW topological invariant (ie $Z_{G,w}(M^4) = Z_{G,u}(M^4)$ for any closed $M^4$).

When the group $G$ has an outer automorphism and if $w$ and $u$ are related by the outer automorphism, then $Z_{G,w}(M^4) = Z_{G,u}(M^4)$. Here I would like to ask if there are examples beyond outer automorphism.

In 3-dimensions, there are some dramatic examples beyond outer automorphism that $Z_{G,w}(M^3) = Z_{H,u}(M^3)$ even when $G\neq H$.

• If $w$ and $u$ differ by a coboundary, then of course this will be true, because the Dijkgraaf-Witten invariants only depend on the cohomology class of the cocycle you twist by. Apart from that, I know of no examples where the theories coincide, and would expect such agreements to be a coincidence. Apr 6 '17 at 21:27