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

References

1. C. Ceballos, F. Santos and G. M. Ziegler, Many non-equivalent realizations of the associahedron, Combinatorica 35(5) (2015) 513–551. Crossref, Web of Science, Google Scholar
2. H.-M. Chen, The Dijkgraaf–Witten invariants of Seifert 3-manifolds with orientable bases, J. Geom. Phys. 108 (2016) 38–48. Crossref, Web of Science, Google Scholar
3. J. Cho, J. Murakami and Y. Yokota, The complex volumes of twist knots, Proc. Amer. Math. Soc. 137(10) (2009) 3533–3541. Crossref, Web of Science, Google Scholar
4. R. H. Crowell and R. Fox, Introduction to Knot Theory, Graduate Texts in Math ematics, Vol. 57 (Springer-Verlag, New York, Heidelberg, Berlin, 1977). Crossref, Google Scholar
5. P. Derbez, Y. Liu and S.-C. Wang, Chern–Simons theory, surface separability, and volumes of 3-manifolds, J. Topol. 8(4) (2015) 933–974. Crossref, Web of Science, Google Scholar
6. T. Dimofte and S. Gukov, Quantum field theory and the volume conjecture, in Interactions Between Hyperbolic Geometry Quantum Topology and Number Theory, Contemporary Mathematics, Vol. 541 (American Mathematical Society, Providence, RI, 2011), pp. 41–67. Crossref, Google Scholar
7. D. S. Freed, Higher algebraic structures and quantization, Commun. Math. Phys. 159(2) (1994) 343–398. Crossref, Web of Science, Google Scholar
8. D. S. Freed, Classical Chern–Simons theory, 1, Adv. Math. 113(2) (1995) 237–303. Crossref, Web of Science, Google Scholar
9. D. S. Freed, Remarks on Chern–Simons theory, Bull. Amer. Math. Soc. 46(2) (2009) 221–254. Crossref, Web of Science, Google Scholar
10. S. Garoufalidis, D. P. Thurston and C. K. Zickert, The complex volume of $SL (n, ℂ)$ -representations of 3-manifolds, Duke Math. J. 164(11) (2015) 2099–2160. Crossref, Web of Science, Google Scholar
11. J. Y. Ham and J. Lee, Explicit formulae for Chern–Simons invariants of the hyperbolic orbifolds of the knot with Conway’s notation $C (2 n, 3)$ , Lett. Math. Phys. 107(3) (2017) 427–437. Crossref, Web of Science, Google Scholar
12. E. Hatakenaka and T. Nosaka, Some topological aspects of 4-fold symmetric quandle invariants of 3-manifolds, Int. J. Math. 23(7) (2012) 1250064. Link, Web of Science, Google Scholar
13. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002). Google Scholar
14. A. Inoue and Y. Kabaya, Quandle homology and complex volume, Geom. Dedicata 171(1) (2014) 265–292. Crossref, Web of Science, Google Scholar
15. P. A. Kirk and E. P. Klassen, Chern–Simons invariants of 3-manifolds and representation spaces of knot groups, Math. Ann. 287 (1990) 343–367. Crossref, Web of Science, Google Scholar
16. T. Kuessner, Fundamental classes of 3-manifold group representations in $SL (4, ℝ)$ , J. Knot Theory Ramifications 26(7) (2017) 1750036. Link, Web of Science, Google Scholar
17. R. Lorentz and D. B. Benson, Deterministic and nondeterministic flow-chart interpretations, J. Comput. System Sci. 27 (1983) 400–433. Crossref, Web of Science, Google Scholar
18. J. Marché, Geometric interpretation of simplicial formulas for the Chern–Simons invariant, Algebr. Geom. Topol. 12(2) (2012) 805–827. Crossref, Web of Science, Google Scholar
19. T. Nosaka, On the fundamental 3-classes of knot group representations, Geom. Dedicata 204 (2020) 1–24. Crossref, Web of Science, Google Scholar
20. V. Turaev, Homotopy Quantum Field Theory, EMS Tracts in Mathematics, Vol. 10 (European Mathematical Society, 2010). Crossref, Google Scholar
21. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121(3) (1989) 351–399. Crossref, Web of Science, Google Scholar
22. C. K. Zickert, The volume and Chern–Simons invariant of a representation, Duke Math. J. 150 (2009) 489–532. Crossref, Web of Science, Google Scholar