3–manifold invariants and periodicity of homology spheres

Gilmer, Patrick M¹ ; Kania-Bartoszynska, Joanna² ; Przytycki, Jozef H³

¹Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
²Department of Mathematics, Boise State University, Boise, ID 83725, USA
³Department of Mathematics, George Washington University, Washington, DC 20052, USA

Algebraic and Geometric Topology, Tome 2 (2002) no. 2, pp. 825-842

Résumé

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.

Détail

DOI : 10.2140/agt.2002.2.825

Keywords: $3$–manifolds, links, group actions, quantum invariants

Bibliographie

[1] C Blanchet, N Habegger, G Masbaum, P Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992) 685

[2] M Boileau, personal communication (2001–2002)

[3] M Boileau, J Porti, Geometrization of 3–orbifolds of cyclic type, Astérisque (2001) 208

[4] G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press (1972)

[5] N Chbili, Les invariants $\theta_p$ des 3–variétés périodiques, Ann. Inst. Fourier (Grenoble) 51 (2001) 1135

[6] Q Chen, T Le, Quantum invariants of periodic links and periodic 3–manifolds, Fund. Math. 184 (2004) 55

[7] T D Cochran, P Melvin, Quantum cyclotomic orders of 3–manifolds, Topology 40 (2001) 95

[8] M Dehn, Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910) 137

[9] R Fenn, C Rourke, On Kirby's calculus of links, Topology 18 (1979) 1

[10] P M Gilmer, Integrality for TQFTs, Duke Math. J. 125 (2004) 389

[11] P M Gilmer, Quantum invariants of periodic three-manifolds, from: "Proceedings of the Kirbyfest (Berkeley, CA, 1998)", Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 157

[12] D L Goldsmith, Symmetric fibered links, from: "Knots, groups, and 3–manifolds (Papers dedicated to the memory of R. H. Fox)", Princeton Univ. Press (1975)

[13] F Hirzebruch, The signature theorem: reminiscences and recreation, from: "Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970)", Princeton Univ. Press (1971)

[14] N Jacobson, Basic algebra II, W. H. Freeman and Company (1989)

[15] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. $(2)$ 126 (1987) 335

[16] L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987)

[17] L H Kauffman, S L Lins, Temperley–Lieb recoupling theory and invariants of 3–manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994)

[18] R Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978) 35

[19] R Kirby, P Melvin, The 3–manifold invariants of Witten and Reshetikhin–Turaev for $\mathrm{sl}(2,\mathbb{C})$, Invent. Math. 105 (1991) 473

[20] R J Lawrence, Witten–Reshetikhin–Turaev invariants of 3–manifolds as holomorphic functions, from: "Geometry and physics (Aarhus, 1995)", Lecture Notes in Pure and Appl. Math. 184, Dekker (1997) 363

[21] G Masbaum, J D Roberts, A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Cambridge Philos. Soc. 121 (1997) 443

[22] W H Meeks Iii, P Scott, Finite group actions on 3–manifolds, Invent. Math. 86 (1986) 287

[23] J W Morgan, The Smith conjecture, from: "The Smith conjecture (New York, 1979)", Pure Appl. Math. 112, Academic Press (1984) 3

[24] H Murakami, Quantum $\mathrm{SO}(3)$–invariants dominate the $\mathrm{ SU}(2)$-invariant of Casson and Walker, Math. Proc. Cambridge Philos. Soc. 117 (1995) 237

[25] K Murasugi, Jones polynomials of periodic links, Pacific J. Math. 131 (1988) 319

[26] H Poincaré, Cinquième Complément à l'Analysis Situs, Rend. Circ. Mat. Palermo 18 (1904) 277

[27] J H Przytycki, On Murasugi's and Traczyk's criteria for periodic links, Math. Ann. 283 (1989) 465

[28] J H Przytycki, An elementary proof of the Traczyk–Yokota criteria for periodic knots, Proc. Amer. Math. Soc. 123 (1995) 1607

[29] J H Przytycki, M V Sokolov, Surgeries on periodic links and homology of periodic 3–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 295

[30] N Reshetikhin, V G Turaev, Invariants of 3–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547

[31] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1976)

[32] M Sakuma, Surgery description of orientation-preserving periodic maps on compact orientable 3–manfolds, Rend. Istit. Mat. Univ. Trieste 32 (2001)

[33] H Seifert, Topologie dreidimensional gefaserte Raume, Acta Math. 60 (1932) 147

[34] P Traczyk, $10_{101}$ has no period 7: a criterion for periodic links, Proc. Amer. Math. Soc. 108 (1990) 845

[35] P Traczyk, Periodic knots and the skein polynomial, Invent. Math. 106 (1991) 73

[36] H Weyl, Invariants, Duke Math. J. 5 (1939) 489

[37] E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351

[38] Y Yokota, The skein polynomial of periodic knots, Math. Ann. 291 (1991) 281

[39] X Zhang, On property I for knots in $S^3$, Trans. Amer. Math. Soc. 339 (1993) 643