Cohomological invariants of representations of 3-manifold groups

Suppose $\mathrm{\Gamma }$ is a discrete group, and $\alpha \in Z^3(B\mathrm{\Gamma };A)$, with $A$ an abelian group. Given a representation $\rho :{\pi }_1(M)\to \mathrm{\Gamma }$, with $M$ a closed 3-manifold, put $F(M,\rho )=⟨(B\rho {)}^{\ast }[\alpha ],[M]⟩$, where $B\rho :M\to B\mathrm{\Gamma }$ is a continuous map inducing $\rho$ which is unique up to homotopy, and $⟨-,-⟩:H^3(M;A)\times H_3(M;\mathbb{Z})\to A$ is the pairing. We extend the definition of $F(M,\rho )$ to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing $F(M,\rho )$ when $M$ is given by a surgery along a link $L\subset S^3$. In particular, the Chern-Simons invariant can be computed this way.