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The Triple-Point Spectrum of Closed Orientable 3-Manifolds

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Abstract

The triple-point numbers and the triple-point spectrum of a closed 3-manifold are topological invariants that give a measure of the complexity of the 3-manifold using the number of triple points of minimal filling Dehn surfaces. Basic properties of these invariants are presented, and the triple-point spectra of \(\mathbb {S}^2\times \mathbb {S}^1\) and \(\mathbb {S}^3\) are computed.

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References

  1. Aitchison, I.R., Matsumotoi, S., Rubinstein, J.H.: Immersed surfaces in cubed manifolds. Asian J. Math. 1(1), 85–95 (1997). https://doi.org/10.4310/AJM.1997.v1.n1.a4

    Article  MathSciNet  MATH  Google Scholar 

  2. Amendola, G.: A local calculus for nullhomotopic filling Dehn spheres. Algebr. Geom. Topol. 9(2), 903–933 (2009). https://doi.org/10.2140/agt.2009.9.903

    Article  MathSciNet  MATH  Google Scholar 

  3. Amendola, G.: A 3-manifold complexity via immersed surfaces. J. Knot Theory Ramif. 19(12), 1549–1569 (2010). https://doi.org/10.1142/S0218216510008558

    Article  MathSciNet  MATH  Google Scholar 

  4. Babson, E.K., Chan, C.S.: Counting faces of cubical spheres modulo two. Discrete Math. 212(3), 169–183 (2000). https://doi.org/10.1016/S0012-365X(99)00285-X. (Combinatorics and applications (Tianjin, 1996))

    Article  MathSciNet  MATH  Google Scholar 

  5. Carter, S., Kamada, S., Saito, M.: Surfaces in 4-space. Encyclopaedia of Mathematical Sciences, vol. 142. Springer, Berlin (2004). (Low-Dimensional Topology, III. MR2060067)

    MATH  Google Scholar 

  6. Fenn, R., Rourke, C.: Nice spines of \(3\)-manifolds, Topology of low-dimensional manifolds, (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math., vol. 722. Springer, Berlin 1979, 31–36 (1977)

  7. Funar, L.: Cubulations, immersions, mappability and a problem of Habegger. Ann. Sci. École Norm. Sup. 32(5), 681–700 (1999). https://doi.org/10.1016/S0012-9593(01)80003-3. (English, with English and French summaries)

    Article  MathSciNet  MATH  Google Scholar 

  8. Haken, W.: Some special presentations of homotopy \(3\)-spheres, Topology Conference, (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973), Springer, Berlin, pp. 97–107. Lecture Notes in Math., Vol. 375 (1974)

  9. Hall Jr., M.: The theory of groups. Chelsea Publishing Co., New York (1976). (Reprinting of the 1968 edition)

    Google Scholar 

  10. Johansson, I.: Über singuläre Elementarflächen und das Dehnsche Lemma. Math. Ann. 110(1), 312–320 (1935). https://doi.org/10.1007/BF01448029. (German)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johansson, I.: Über singuläre Elementarflächen und das Dehnsche Lemma II. Math. Ann. 115(1), 658–669 (1938). https://doi.org/10.1007/BF01448964. (German)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lozano, Á., Vigara, R.: On the subadditivity of Montesinos complexity of closed orientable 3-manifolds. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109(2), 267–279 (2015). https://doi.org/10.1007/s13398-014-0179-1

    Article  MathSciNet  MATH  Google Scholar 

  13. Montesinos-Amilibia, J.M.: Representing 3-manifolds by Dehn spheres, Mathematical contributions: volume in honor of Professor Joaquín Arregui Fernández (Spanish), Homen. Univ. Complut., Editorial Complutense, Madrid, pp. 239–247 (2000)

  14. Papakyriakopoulos, C.D.: On Dehn’s lemma and the asphericity of knots. Ann. Math. (2) 66, 1–26 (1957). https://doi.org/10.2307/1970113

    Article  MathSciNet  MATH  Google Scholar 

  15. Shima, A.: Immersions from the \(2\)-sphere to the \(3\)-sphere with only two triple points. Sūrikaisekikenkyūsho Kōkyūroku 1006, 146–160 (1997). (Topology of real singularities and related topics (Japanese) (Kyoto, 1997))

    MathSciNet  MATH  Google Scholar 

  16. Vigara, R.: A set of moves for Johansson representation of 3-manifolds. Fund. Math. 190, 245–288 (2006). https://doi.org/10.4064/fm190-0-10

    Article  MathSciNet  MATH  Google Scholar 

  17. Vigara, R., Lozano, Á.: Representing 3-manifolds by filling Dehn surfaces, Series on Knots and Everything, vol. 58, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2016)

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Authors and Affiliations

  1. Centro Universitario de la Defensa Zaragoza, Academia General Militar Carretera de Huesca s/n., 50090, Zaragoza, Spain

    Álvaro Lozano Rojo & Rubén Vigara

  2. IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Álvaro Lozano Rojo & Rubén Vigara

Authors
  1. Álvaro Lozano Rojo
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  2. Rubén Vigara
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Correspondence to Álvaro Lozano Rojo.

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Partially supported by Gobierno de Aragon (Ref. Grupo Algebra y Geometría) co-funded by Feder 2014-2020 “Construyendo Europa desde Aragón”, and MINECO grants MTM2013-45710-C2-1-P, MTM2013-46337-C2-2-P, MTM2016-77642-C2-2-P and MTM2016-76868-C2-2-P.

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Lozano Rojo, Á., Vigara, R. The Triple-Point Spectrum of Closed Orientable 3-Manifolds. Mediterr. J. Math. 16, 71 (2019). https://doi.org/10.1007/s00009-019-1340-z

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  • DOI: https://doi.org/10.1007/s00009-019-1340-z

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