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On the fundamental 3-classes of knot group representations. (English) Zbl 1439.57024

Taking a link exterior and the peripheral system as a model, a topological study of a more general connected \(3\)-manifold \(M\) whose boundary is a union of tori aims at investigating a pairing given by the following data: a group representation \(\pi_1 M \to G\) mapping the image of \(\pi_1 \partial M\) to a subgroup \(K\), a \({\mathbb Z}[G]\)-module \(A\), and a \(3\)-cocycle \(\theta\) from \(H^3(G,K;A)\); then \(\theta\) and a pushforward \(f_*[M,\partial M] \in H_3(M,\partial M; {\mathbb Z})\) are paired to \(\langle \theta, f_*[M,\partial M] \rangle \in A_G = H_0(G;A)\). This pairing occurs in the context of the volume and the Chern-Simons invariant of a hyperbolic manifold as well as of the Dijkgraaf-Witten invariant. So far, the pairing was thought inaccessible to explicit computations. Here, for rather general types of knots, a calculable description is presented that starts out with geometric configurations such as diagrams. Based on key ideas from [Geom. Dedicata 171, 265–292 (2014; Zbl 1300.57012)] by A. Inoue and Y. Kabaya the author uses quandles as a bridge to knot theory. So he applies the findings of D. Joyce [J. Pure Appl. Algebra 23, 37–65 (1982; Zbl 0474.57003)] that the knot quandle classifies knots, which in turn rests upon Waldhausen’s deep result on knot classification by knot group plus peripheral system.

MSC:

57K10 Knot theory
57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)
57K31 Invariants of 3-manifolds (including skein modules, character varieties)
57K32 Hyperbolic 3-manifolds
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
20C99 Representation theory of groups
20J06 Cohomology of groups
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References:

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