Cohomological invariants of representations of 3-manifold groups
Abstract
Suppose ΓΓ is a discrete group, and α∈Z3(BΓ;A)α∈Z3(BΓ;A), with AA an abelian group. Given a representation ρ:π1(M)→Γρ:π1(M)→Γ, with MM a closed 3-manifold, put F(M,ρ)=〈(Bρ)∗[α],[M]〉F(M,ρ)=⟨(Bρ)∗[α],[M]⟩, where Bρ:M→BΓBρ:M→BΓ is a continuous map inducing ρρ which is unique up to homotopy, and 〈−,−〉:H3(M;A)×H3(M;ℤ)→A is the pairing. We extend the definition of F(M,ρ) to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing F(M,ρ) when M is given by a surgery along a link L⊂S3. In particular, the Chern–Simons invariant can be computed this way.