A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients:
- An oriented, (assumed here to be smooth) manifold $M^n$
- A finite group $G$ (and a field, chosen to be $\mathbb{C}$ here)
- An $n$-th cohomology class $[\omega] \in H^n(G, U(1))$ (where $U(1)$ is considered as a discrete group here)
- A homomorphism $\phi\colon \pi_1(M) \to G$
There is a canonical (up to homotopy) map $c\colon M \to B\pi_1(M)$. We can then abstractly construct the cohomology class $c^* \phi^* ([\omega]) \in H^n(M, U(1))$.
I'm interested describing this cohomology class very concretely. Assume I have given:
- $M$ as a handle decomposition (and consequently we consider Morse cohomology, where the $k$-th grade of the complex is generated by the $k$-handles)
- $\phi$ as an assignment of group elements to each 1-handle of $M$ (satisfying the relations for 2-handles)
- $[\omega]$ represented through a concrete cocycle $\omega\colon G^n \to U(1)$, i.e. a cocycle in the simplicial cohomology of $BG$
How can I explicitly pull back $[\omega]$ along $\phi \circ c$?
Some remarks:
- A handle decomposition of $M$ also gives a CW-complex, I think. So $c\colon M \to B\pi_1(M)$ can be chosen canonically as a cellular map via the Postnikov construction, and it's the pullback on cohomology can be derived from the pullback on the level of complexes.
- The trouble seems to be that $\phi$ isn't obviously cellular on higher cells if we describe $BG$ as a simplicial complex (which we might have to at some point because $[\omega]$ is given that way).
