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Lower bounds for regular genus and gem-complexity of PL 4-manifolds

  • Biplab Basak and Maria Rita Casali EMAIL logo
Published/Copyright: August 11, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 4

Abstract

Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely, gem-complexity and regular genus. In the present paper we prove that for any closed connected PL 4-manifold M, its gem-complexity k⁢(M) and its regular genus 𝒢⁢(M) satisfy

k⁢(M)≥3⁢χ⁢(M)+10⁢m-6 and 𝒢⁢(M)≥2⁢χ⁢(M)+5⁢m-4,

where r⁢k⁢(π1⁢(M))=m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of “standard type”, involved in existing crystallization catalogs, and their connected sums.

Keywords: PL 4-manifold; pseudo-triangulation; crystallization; regular genus; gem-complexity; semi-simple crystallization
MSC 2010: 57Q15; 57Q05; 57N13; 05C15

Communicated by Frederick R. Cohen

Funding statement: The first author is supported by CSIR, India for SPM Fellowship (400149/SPMF 2012 Award EMR 1) and the UGC Centre for Advanced Studies. The second author is supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INDAM) and by M.I.U.R. of Italy (project “Strutture Geometriche, Combinatoria e loro Applicazioni”).

Acknowledgements

The authors express their gratitude to Prof. Basudeb Datta and Dr. Jonathan Spreer for helpful comments.

References

[1] P. Bandieri, P. Cristofori and C. Gagliardi, Nonorientable 3-manifolds admitting colored triangulations with at most 30 tetrahedra, J. Knot Theory Ramifications 18 (2009), 381–395. 10.1142/S0218216509006975Search in Google Scholar

[2] B. Basak and B. Datta, Minimal crystallizations of 3-manifolds, Electron. J Combin. 21 (2014), no. 1, Research Paper P1.61. 10.37236/3956Search in Google Scholar

[3] B. Basak and J. Spreer, Simple crystallizations of 4-manifolds, Adv. Geom. 16 (2016), no. 1, 111–130. 10.1515/advgeom-2015-0043Search in Google Scholar

[4] A. Björner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984), 7–16. 10.1016/S0195-6698(84)80012-8Search in Google Scholar

[5] J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, New York, 2008. 10.1007/978-1-84628-970-5Search in Google Scholar

[6] M. R. Casali, An infinite class of bounded 4-manifolds having regular genus three, Boll. Unione Mat. Ital. A 10 (1996), no. 7, 279–303. Search in Google Scholar

[7] M. R. Casali, Classifying PL 5-manifolds by regular genus: The boundary case, Canad. J. Math. 49 (1997), 193–211. 10.4153/CJM-1997-010-3Search in Google Scholar

[8] M. R. Casali, Classification of non-orientable 3-manifolds admitting decompositions into ≤26 colored tetrahedra, Acta Appl. Math. 54 (1998), 75–97. 10.1023/A:1006014705085Search in Google Scholar

[9] M. R. Casali, Computing Matveev’s complexity of non-orientable 3-manifolds via crystallization theory, Topology Appl. 144 (2004), 201–209. 10.1016/j.topol.2004.04.010Search in Google Scholar

[10] M. R. Casali and P. Cristofori, Computing Matveev’s complexity via crystallization theory: The orientable case, Acta Appl. Math. 92 (2006), no. 2, 113–123. 10.1007/s10440-006-9065-ySearch in Google Scholar

[11] M. R. Casali and P. Cristofori, A catalogue of orientable 3-manifolds triangulated by 30 colored tetrahedra, J. Knot Theory Ramifications 17 (2008), 1–23. 10.1142/S0218216508006312Search in Google Scholar

[12] M. R. Casali and P. Cristofori, Cataloguing PL 4-manifolds by gem-complexity, Electron. J. Combin. 22 (2015), no. 4, Article ID #P4.25. 10.37236/4749Search in Google Scholar

[13] M. R. Casali, P. Cristofori and C. Gagliardi, PL 4-manifolds admitting simple crystallizations: framed links and regular genus, J. Knot Theory Ramifications 25 (2016), no. 4, 1–14. 10.1142/S021821651650005XSearch in Google Scholar

[14] M. R. Casali and C. Gagliardi, Classifying PL 5-manifolds up to regular genus seven, Proc. Amer. Math. Soc. 120 (1994), 275–283. 10.1090/S0002-9939-1994-1205484-4Search in Google Scholar

[15] M. R. Casali and L. Malagoli, Handle-decompositions of PL 4-manifolds, Cah. Topol. Géom. Différ. Catég. 38 (1997), 141–160. Search in Google Scholar

[16] A. Cavicchioli, On the genus of smooth 4-manifolds, Trans. Amer. Math. Soc. 31 (1999), 203–214. 10.1090/S0002-9947-1992-1034659-5Search in Google Scholar

[17] A. Cavicchioli and M. Meschiari, On classification of 4-manifolds according to genus, Cah. Topol. Géom. Différ. Catég. 34 (1993), 37–56. Search in Google Scholar

[18] M. Ferri and C. Gagliardi, The only genus zero n-manifold is 𝕊n, Proc. Amer. Math. Soc. 85 (1982), 638–642. 10.1090/S0002-9939-1982-0660620-5Search in Google Scholar

[19] M. Ferri, C. Gagliardi and L. Grasselli, A graph-theoretic representation of PL-manifolds – A survey on crystallizations, Aequationes Math. 31 (1986), 121–141. 10.1007/BF02188181Search in Google Scholar

[20] C. Gagliardi, Extending the concept of genus to dimension n, Proc. Amer. Math. Soc. 81 (1981), 473–481. 10.2307/2043490Search in Google Scholar

[21] C. Gagliardi and L. Grasselli, Representing products of polyhedra by products of edge-colored graphs, J. Graph Theory 17 (1993), 549–579. 10.1002/jgt.3190170502Search in Google Scholar

[22] S. Lins, Gems, Computers and Attractors for 3-Manifolds, Ser. Knots Everything 5, World Scientific, Hackensack, 1995. 10.1142/2490Search in Google Scholar

[23] M. Pezzana, Sulla struttura topologica delle varietà compatte, Atti Sem. Mat. Fis. Univ. Modena 23 (1974), 269–277. Search in Google Scholar

[24] F. Spaggiari, On the genus of ℝ⁢ℙ3×𝕊1, Collect. Math. 50 (1999), no. 3, 229–241. Search in Google Scholar

[25] C. T. C. Wall, On simply-connected 4-manifolds, J. Lond. Math. Soc. (2) 39 (1964), 141–149. 10.1112/jlms/s1-39.1.141Search in Google Scholar

Received: 2015-5-3
Published Online: 2016-8-11
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
https://doi.org/10.1515/forum-2015-0080
Keywords for this article
PL 4-manifold; pseudo-triangulation; crystallization; regular genus; gem-complexity; semi-simple crystallization

Articles in the same Issue

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
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