# Cohomological invariants of representations of 3-manifold groups

## TL;DR

This paper introduces a cohomological invariant for representations of 3-manifold groups, extending its definition to manifolds with corners, and provides a practical computation method, including for the Chern-Simons invariant.

## Contribution

It extends the invariant to manifolds with corners and develops a gluing law, enabling practical computation from link surgery presentations.

## Key findings

- Defined a cohomological invariant for 3-manifold group representations.
- Extended the invariant to manifolds with corners.
- Provided a method to compute the invariant via link surgery, including the Chern-Simons invariant.

## 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 $F(M,\rho)=\langle(B\rho)^\ast[\alpha],[M]\rangle$, where $B\rho:M\to B\Gamma$ is a continuous map inducing $\rho$ which is unique up to homotopy, and $\langle-,-\rangle:H^3(M;A)\times H_3(M;\mathbb{Z})\to A$ 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.