group cohomology of 3-manifolds

Question

If $M$ is a closed aspherical 3-manifold with first fundamental group $G$, then cohomology groups of $G$ and $M$ are isomorphic because $M$ is a Eilenberg-MacLane space $K(G,1)$. In particular there should be a description of fundamental cohomology class of $M$, $[M]$, in terms of group cohomology, i.e. in inhomogeneous coordinates for any triple $(g_0 , g_1 , g_2)$ of elements of $G$, $[M]$ assigns a number to this triple.

Question: How to describe these numbers.

P.S. : references for group cohomology in a geometric language would be appreciated.

P.S. : I'm interested in 3-manifolds that admit a taut foliation so their universal cover is $\mathbb R^3$, so they are aspherical, in case this extra assumption helps.

Answer 1

I suggest looking up a few sources, which will allow you to piece together your own proof.

1. The proof of Poincare duality in the book of Bott-Tu, with particular emphasis on the construction of a De Rham cocycle representing $[M]$.
2. The construction of a cochain map representing the isomorphism between De Rham cohomology and singular cohomology with real coefficients. Again, Bott-Tu is a good source.
3. The construction of a cochain map representing the universal coefficients isomorphism between $H^n(M;\mathbb{Z}) \otimes \mathbb{R} \to H^n(M;\mathbb{R})$. For this, any algebraic topology textbook would be good.
4. The construction of a cochain map representing the isomorphism between $H^n(\pi_1 M;\mathbb{Z})$ and $H^n(M;\mathbb{Z})$. For this, Brown's book on "Cohomology of Groups" would be good. This is where you'll need that $M$ is aspherical.

String those isomorphisms together and you'll have your group cocycle.