Algebraic & Geometric Topology

3–manifold invariants and periodicity of homology spheres

Patrick M Gilmer, Joanna Kania-Bartoszynska, and Jozef H Przytycki

Full-text: Open access

Abstract

We show how the periodicity of a homology sphere is reflected in the Reshetikhin–Turaev–Witten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.

Article information

Source
Algebr. Geom. Topol., Volume 2, Number 2 (2002), 825-842.

Dates
Received: 19 November 2001
Revised: 3 September 2002
Accepted: 6 September 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882743

Digital Object Identifier
doi:10.2140/agt.2002.2.825

Mathematical Reviews number (MathSciNet)
MR1936972

Zentralblatt MATH identifier
1008.57013

Subjects
Primary: 57M60: Group actions in low dimensions 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57R56: Topological quantum field theories 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]

Keywords
$3$–manifolds links group actions quantum invariants

Citation

Gilmer, Patrick M; Kania-Bartoszynska, Joanna; Przytycki, Jozef H. 3–manifold invariants and periodicity of homology spheres. Algebr. Geom. Topol. 2 (2002), no. 2, 825--842. doi:10.2140/agt.2002.2.825. https://projecteuclid.org/euclid.agt/1513882743


Export citation

References

  • C. Blanchet, N. Habegger, G. Masbaum, P. Vogel: Three manifold invariants derived from the Kauffman bracket, Topology 31 (1992), 685-699
  • C. Blanchet, N. Habegger, G. Masbaum, P. Vogel: Topological quantum field theories derived from the Kauffman bracket, Topology \bf34 (1995), 883-927
  • M. Boileau, Personal communication, Sept. 2001, March 2002.
  • M. Boileau, J. Porti: Geometrization of 3-orbifolds of cyclic type, Asterisque 272, (2001).
  • G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972.
  • N. Chbili : Les invariants $\theta_p$ des 3-varietes periodiques, preprint (2001).
  • Q. Chen and T. Le: Quantum invariants of periodic links and periodic 3-manifolds, preprint (2002).
  • T. D Cochran, P. Melvin: Quantum cyclotomic orders of 3-manifolds, Topology, 40 (2001), no. 1, 95–125.
  • M. Dehn: Über die Topologie des dreidimensionalen Raumes, Math. Annalen. 69, 137-168 (1910).
  • R. Fenn, C. Rourke: On Kirby's calculus of links, Topology 18, 1–15 (1979).
  • P. M. Gilmer: Integrality for TQFTS.
  • P. M. Gilmer: Quantum invariants of periodic three-manifolds, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., 2, 157–175, (1999)
  • D. L. Goldsmith: Symmetric fibered links, in: Knots, groups and 3-manifolds ed.: L. P. Neuwirth, Princeton University Press, 1975.
  • F. Hirzebruch: The signature theorem: reminiscences and recreation, Prospects of Math, Ann. of Math Stud. 70, Princeton University Press, 3–31 (1971).
  • N. Jacobson, Basic Algebra II, second edition, W.H. Freeman (1989)
  • V. Jones: Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126, 335–388 (1987).
  • L. Kauffman: On Knots, Annals of Mathematics Studies 115, Princeton University Press, 1987.
  • L. Kauffman and S. Lins : Temperley-Lieb Recoupling Theory and Invariants of 3-manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994).
  • R. C. Kirby: A calculus for framed links in $S^3$, Invent. Math. 45, 35–56 (1978).
  • R. C. Kirby, P. Melvin: The 3-manifold invariants of Witten and Reshetikhin-Turaev for $sl(2,{\bf C})$. Invent. Math. 105, 473–545 (1991).
  • R. J. Lawrence: Witten-Reshetikhin-Turaev invariants of $3$-manifolds as holomorphic functions, Geometry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., 184, 363–377, Dekker, New York, (1997).
  • G. Masbaum, J. Roberts: A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Cambridge Philos. Soc., 121, 443–454 (1997).
  • W. M. Meeks III, P. Scott,: Finite group actions on $3$-manifolds, Invent. Math. 86(2), 287–346 (1986).
  • J. W. Morgan, H. Bass, eds: The Smith conjecture, Pure Appl. Math. 112, New York Academic Press, 1984.
  • H. Murakami: Quantum $SO(3)$-invariants dominate the $SU(2)$-invariant of Casson and Walker, Math. Proc. Cambridge Philos. Soc., 117, 237–249 (1995).
  • K. Murasugi: Jones polynomials of periodic links, Pacific J. Math. 131, 319–329 (1988).
  • H. Poincaré: Cinquième Complément à l'Analysis Situs, Rend. Circ. Mat. Palermo 18, 277-308 (1904).
  • J. H. Przytycki: On Murasugi's and Traczyk's criteria for periodic links, Math. Ann. 283, 465–478 (1989).
  • J. H. Przytycki: An elementary proof of the Traczyk-Yokota criteria for periodic knots, Proc. Amer. Math. Soc. 123, 1607–1611 (1995).
  • J. H. Przytycki, M. Sokolov : Surgeries on periodic links and homology of periodic 3-manifolds. Math. Proc. Cambridge Philos. Soc. \bf131, 295–307 (2001).
  • N. Yu. Reshetikhin, V. G. Turaev: Invariants of 3-manifolds via Link Polynomials and Quantum Groups, Invent. Math. 103, 547–597 (1991).
  • D. Rolfsen: Knots and Links, Publish or Perish, (1976).
  • M. Sakuma: Surgery description of orientation-preserving periodic maps on compact orientable 3-manifolds, Rend. Istit. Mat. Univ. Trieste XXXII, 1-11 (2000).
  • H. Seifert : Topologie dreidimensional gefaserte Raume, Acta Math 60, 147-238 (1932).
  • P. Traczyk: $10_{101}$ has no period $7$: a criterion for periodicity of links, Proc. Amer. Math. Soc. 180, 845–846 (1990).
  • P. Traczyk: Periodic links and the skein polynomial, Invent. Math. 106, 73–84 (1991).
  • Herman Weyl: Invariants, Duke Mathematical Journal 5, 489–502 (1939).
  • E. Witten: Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, 351–399 (1989).
  • Y. Yokota: The skein polynomial of periodic links, Math. Ann. 291, 281–291 (1991).
  • X. Zhang: On property I for knots in $S^3$, Trans. AMS 339, 643–657 (1993).