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Cohomological invariants of representations of 3-manifold groups

Haimiao Chen

Department of Mathematics, Beijing Technology and Business University, 11# Fucheng Road, Haidian District, Beijing, P. R. China

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https://doi.org/10.1142/S0218216520430038Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue: Moscow-Beijing Conference on Topology
Guest Editors: Z. Wan, S. Ren and V. O. Manturov

Abstract

Suppose $Γ$ $Γ$ is a discrete group, and $α \in Z^{3} (B Γ; A)$ $α \in Z^{3} (B Γ; A)$ , with $A$ $A$ an abelian group. Given a representation $ρ : π_{1} (M) \to Γ$ $ρ : π_{1} (M) \to Γ$ , with $M$ $M$ a closed 3-manifold, put $F (M, ρ) = ⟨ {(B ρ)}^{*} [α], [M] ⟩$ $F (M, ρ) = 〈 {(B ρ)}^{*} [α], [M] 〉$ , where $B ρ : M \to B Γ$ $B ρ : M \to B Γ$ is a continuous map inducing $ρ$ $ρ$ which is unique up to homotopy, and $〈 -, - 〉 : H^{3} (M; A) \times H_{3} (M; ℤ) \to A$ 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.

Keywords:

AMSC: 57K31

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