Cohomological invariants of representations of 3-manifold groups

Abstract

Suppose $\Gamma$ is a discrete group, and $\alpha \in Z^3(B\Gamma ;A)$, with $A$ an abelian group. Given a representation $\rho :{\pi }_1(M)\to \Gamma$, with $M$ a closed 3-manifold, put $F(M,\rho )=⟨(B\rho {)}^{\ast }[\alpha ],[M]⟩$, where $B\rho :M\to B\Gamma$ 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.

18 pages, 19 figures

References in corpus (2)

• Chern--Simons theory, surface separability, and volumes of 3-manifolds
• Explicit formulae for Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation $C(2n,3)$