Algebraic & Geometric Topology

Spin structures on $3$–manifolds via arbitrary triangulations

Riccardo Benedetti and Petronio Carlo

Full-text: Open access

Abstract

Let M be an oriented compact 3–manifold and let T be a (loose) triangulation of M with ideal vertices at the components of M and possibly internal vertices. We show that any spin structure s on M can be encoded by extra combinatorial structures on T. We then analyze how to change these extra structures on T, and T itself, without changing s, thereby getting a combinatorial realization, in the usual “objects/moves” sense, of the set of all pairs (M,s). Our moves have a local nature, except one, that has a global flavour but is explicitly described anyway. We also provide an alternative approach where the global move is replaced by simultaneous local ones.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 2 (2014), 1005-1054.

Dates
Received: 16 April 2013
Revised: 9 September 2013
Accepted: 15 September 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715856

Digital Object Identifier
doi:10.2140/agt.2014.14.1005

Mathematical Reviews number (MathSciNet)
MR3180826

Zentralblatt MATH identifier
1302.57061

Subjects
Primary: 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M20: Two-dimensional complexes

Keywords
$3$–manifold spin structure triangulation spine

Citation

Benedetti, Riccardo; Carlo, Petronio. Spin structures on $3$–manifolds via arbitrary triangulations. Algebr. Geom. Topol. 14 (2014), no. 2, 1005--1054. doi:10.2140/agt.2014.14.1005. https://projecteuclid.org/euclid.agt/1513715856


Export citation

References

  • G Amendola, R Benedetti, F Costantino, C Petronio, Branched spines of $3$–manifolds and Reidemeister torsion of Euler structures, Rend. Istit. Mat. Univ. Trieste 32 (2001) 1–33
  • S Baseilhac, R Benedetti, Analytic families of quantum hyperbolic invariants and their asymptotical behaviour, I
  • S Baseilhac, R Benedetti, Quantum hyperbolic invariants of $3$–manifolds with ${\rm PSL}(2,\mathbb C)$–characters, Topology 43 (2004) 1373–1423
  • S Baseilhac, R Benedetti, Classical and quantum dilogarithmic invariants of flat ${\rm PSL}(2,\mathbb C)$–bundles over $3$–manifolds, Geom. Topol. 9 (2005) 493–569
  • R Benedetti, C Petronio, A finite graphic calculus for $3$–manifolds, Manuscripta Math. 88 (1995) 291–310
  • R Benedetti, C Petronio, Branched standard spines of $3$–manifolds, Lecture Notes in Mathematics 1653, Springer, Berlin (1997)
  • R Benedetti, C Petronio, Reidemeister–Turaev torsion of $3$–dimensional Euler structures with simple boundary tangency and pseudo-Legendrian knots, Manuscripta Math. 106 (2001) 13–61
  • C Blanchet, Invariants on three-manifolds with spin structure, Comment. Math. Helv. 67 (1992) 406–427
  • J Christy, Branched surfaces and attractors, I: Dynamic branched surfaces, Trans. Amer. Math. Soc. 336 (1993) 759–784
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • I Ishii, Flows and spines, Tokyo J. Math. 9 (1986) 505–525
  • F Luo, private communication (2011–2013)
  • F Luo, J-M Schlenker, Volume maximization and the extended hyperbolic space, Proc. Amer. Math. Soc. 140 (2012) 1053–1068
  • F Luo, S Tillmann, T Yang, Thurston's spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic $3$–manifolds, Proc. Amer. Math. Soc. 141 (2013) 335–350
  • S V Matveev, Transformations of special spines, and the Zeeman conjecture, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 1104–1116, 1119 In Russian; translated in Math. USSR-Izv. 31 (1988) 423–434
  • S V Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990) 101–130
  • R Piergallini, Standard moves for standard polyhedra and spines, Rend. Circ. Mat. Palermo Suppl. (1988) 391–414
  • N Reshetikhin, V G Turaev, Invariants of $3$–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547–597
  • J Roberts, Refined state-sum invariants of $3$– and $4$–manifolds, from: “Geometric topology”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 217–234
  • V G Turaev, O Y Viro, State sum invariants of $3$–manifolds and quantum $6j$–symbols, Topology 31 (1992) 865–902
  • R F Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. (1974) 169–203