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Cohomological invariants of representations of 3-manifold groups

Haimiao Chen

Abstract

Suppose $Γ$ is a discrete group, and $α\in Z^3(BΓ;A)$, with $A$ an abelian group. Given a representation $ρ:π_1(M)\toΓ$, with $M$ a closed 3-manifold, put $F(M,ρ)=\langle(Bρ)^\ast[α],[M]\rangle$, where $Bρ:M\to BΓ$ is a continuous map inducing $ρ$ 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 $F(M,ρ)$ 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 $L\subset S^3$. In particular, the Chern-Simons invariant can be computed this way.

Cohomological invariants of representations of 3-manifold groups

Abstract

Suppose is a discrete group, and , with an abelian group. Given a representation , with a closed 3-manifold, put , where is a continuous map inducing which is unique up to homotopy, and is the pairing. We extend the definition of to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing when is given by a surgery along a link . In particular, the Chern-Simons invariant can be computed this way.

Paper Structure

This paper contains 13 sections, 2 theorems, 52 equations, 19 figures.

Table of Contents

  1. Introduction
  2. General framework
  3. Preparation
  4. Extending to manifolds with boundary
  5. Fundamental cycle
  6. Cycle-cocycle calculus
  7. Pair of pants $P=\Sigma_{0,3}$
  8. Disk with at least 3 holes removed
  9. Cylinder $C$
  10. Torus $\mathbb{T}^2$
  11. A practical method for computations
  12. Link complements
  13. Closed 3-manifolds with a surgery presentation

Key Result

Lemma 2.4

Suppose $\partial X=Y'\sqcup Y_1\sqcup -Y_2$, and $\varphi:Y_1\stackrel{\approx}\longrightarrow Y_2$ is an orientation-preserving homeomorphism. Let $X^\varphi=X/\{y\sim\varphi(y), y\in Y_1\}$, i.e. the manifold obtained from gluing $X$ along $\varphi$, so that $\partial X^\varphi=Y'$; let ${\rm gl}

Figures (19)

  • Figure 1: (a) An s-triangulation for a square; (b) a 3-simplex
  • Figure 2: The identity (\ref{['eq:calculus-square']})
  • Figure 3: The reason for (\ref{['eq:identity2']})
  • Figure 4: (a) The standard s-triangulation $\xi^{\rm st}_P$ of the pair of pants; (b) depicted in gray is the image of $\xi^{\rm st}_P$ under the twist $\varphi$
  • Figure 5: (a) $\xi_1,\xi_2\in[\Sigma_{0,4}]$; (b) the white part is $[0\stackrel{a}{\frown}0\stackrel{b}{\frown}0\stackrel{c}{\frown}0]$
  • ...and 14 more figures

Theorems & Definitions (10)

  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: Gluing law
  • proof
  • Remark 2.5
  • Remark 2.7
  • Lemma 2.10
  • proof
  • Example 3.3
  • Example 3.4