Skip to main content
SpringerLink
Log in

On reduced complexity of closed piecewise linear 5-manifolds

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5-manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of \(\mathbb {S}^2 \times \mathbb {S}^3\) among closed connected spin PL 5-manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. B. Bagchi-B, Datta, on polytopal upper bound spheres. Mathematika 50, 493–496 (2013)

    Article  Google Scholar 

  2. P. Bandieri, M.R. Casali, C. Gagliardi, Representing manifolds by crystallization theory: foundations, improvements and related results, Atti Sem. Mat. Fis. Univ. Modena. IL, 283–337 (2001)

    MathSciNet  MATH  Google Scholar 

  3. D. Barden, Simply connected five-manifolds. Ann. Math. 82(2), 365–385 (1965)

    Article  MathSciNet  Google Scholar 

  4. J. Bracho, L. Montejano, The combinatorics of colored triangulations of manifolds. Geom. Dedicata 22, 303–328 (1987)

    Article  MathSciNet  Google Scholar 

  5. M.R. Casali, Average order of coloured triangulations: the general case. Osaka J. Math. 35, 249–262 (1998)

    MathSciNet  MATH  Google Scholar 

  6. M.R. Casali, P. Cristofori, A catalogue of orientable \(3\)-manifolds triangulated by \(30\) coloured tetrahedra. J. Knot Theory Ram. 17, 1–23 (2008)

    Article  Google Scholar 

  7. M.R. Casali, P. Cristofori, Cataloguing PL \(4\)-manifolds by gem-complexity. Electron. J. Combin. 22(4), 1–25 (2015)

    Article  MathSciNet  Google Scholar 

  8. M.R. Casali-C, C. Gagliardi, Classifying PL \(5\)-manifolds up to regular genus seven. Proceed. Amer. Math. Soc. 120(1), 275–283 (1994)

    Article  MathSciNet  Google Scholar 

  9. A. Cavicchioli, A combinatorial characterization of \(\mathbb{S}^3 \times \mathbb{S}^1\) among closed 4-manifolds. Proceed. Amer. Math. Soc. 105, 1008–1014 (1989)

    MathSciNet  MATH  Google Scholar 

  10. A. Cavicchioli, F. Hegenbarth, On the determination of PL manifolds by handles of lower dimension. Topol. Appl. 53, 111–118 (1993)

    Article  MathSciNet  Google Scholar 

  11. A. Cavicchioli, F. Spaggiari, On the topological structure of compact 5-manifolds. Comment. Math. Univ. Carol. 34(3), 513–524 (1993)

    MathSciNet  MATH  Google Scholar 

  12. A. Cavicchioli, F. Spaggiari, On the genus of real projective spaces. Arch. Math. 89, 570–576 (2007)

    Article  MathSciNet  Google Scholar 

  13. A. Cavicchioli, F. Spaggiari, Classifying combinatorial \(4\)-manifolds up to complexity. Bol. Soc. Mat. Mex. 14(3), 303–319 (2008)

    MathSciNet  MATH  Google Scholar 

  14. A. Cavicchioli, F. Spaggiari, The combinatorics of piecewise linear manifolds by colored graphs. Int. Math. Forum 7(8), 339–368 (2012)

    MathSciNet  MATH  Google Scholar 

  15. A. Cavicchioli, D. Repovš-, A.B. Skopenkov, Open problems on graphs arising from geometric topology. Topol. Appl. 84, 207–226 (1998)

    Article  MathSciNet  Google Scholar 

  16. F. Costantino, Complexity of 4-manifolds. Exp. Math. 15(2), 237–249 (2006)

    Article  MathSciNet  Google Scholar 

  17. C. Gagliardi, L. Grasselli, Representing products of polyhedra by products of edge-colored graphs. J. Graph Theory 17(5), 549–579 (1993)

    Article  MathSciNet  Google Scholar 

  18. B. Grunbaum, Convex polytopes, in Graduate Texts in Math, 2nd edn., ed. by V. Kaibel, V. Klee, G.M. Ziegler (Springer, Berlin, 1967)

    Google Scholar 

  19. F. Harary, Graph Theory, (Addison-Wesley Publ. Company, Reading, Massachusetts, 1969)

  20. P.J. Hilton, S. Wylie, An Introduction to Algebraic Topology- Homology Theory (Cambridge University Press, Cambridge, 1960)

    Book  Google Scholar 

  21. V. Klee, A combinatorial analogue of Poincaré duality theorem. Can. J. Math. 16, 517–531 (1964)

    Article  Google Scholar 

  22. W. Kühnel, Tight Polyhedral Submanifolds and Tight Triangulations Lect. Notes in Math. (Springer, Berlin, 1995)

    Book  Google Scholar 

  23. W. Kühnel, F. Lutz, A census of tight triangulations. Periodica Math. Hungarica 39, 161–183 (1999)

    Article  MathSciNet  Google Scholar 

  24. S. Lins, Gems, Computers and Attractors for 3-manifolds Series on Knots and Everything (World Scientific Publ., Singapore, 1995)

    MATH  Google Scholar 

  25. F. Luo, R. Stong, Combinatorics of triangulations of \(3\)-manifolds. Trans. Amer. Math. Soc. 337(2), 891–906 (1993)

    MathSciNet  MATH  Google Scholar 

  26. F.H. Lutz, The manifold page, 1999–2005. http://page.math.tu-berlin.de/\(\sim \) lutz/stellar/vertex-transitive-triangulations.html

  27. F. H. Lutz, Triangulated Manifolds with Few Vertices: Combinatorial Manifolds, 2005. arXiv: math.CO/0506372

  28. F.H. Lutz, Combinatorial 3-manifolds with 10 vertices. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 49(1), 97–106 (2008)

    MathSciNet  MATH  Google Scholar 

  29. B. Martelli, Complexity of PL manifolds, Algebraic & Geometric. Topology 10, 1107–1164 (2010)

    MathSciNet  MATH  Google Scholar 

  30. B. Martelli, C. Petronio, Three-manifolds having complexity at most 9. Exp. Math. 10, 207–237 (2004)

    Article  MathSciNet  Google Scholar 

  31. S. Matveev, Complexity theory of 3-dimensional manifolds. Acta Appl. Math. 19, 101–130 (1990)

    MathSciNet  MATH  Google Scholar 

  32. P. McMullen, D.W. Walkup, A generalized lower bound conjecture for simplicial polytopes. Mathematika 18, 264–273 (1971)

    Article  MathSciNet  Google Scholar 

  33. I. Novik, E. Swartz, Socles of buchsbaum modules, complexes and posets. Adv. Math. 222, 2059–2084 (2009)

    Article  MathSciNet  Google Scholar 

  34. G. Perelman, Ricci flow with surgery on three manifolds, arXiv: math. DG/0303109 (2003)

  35. G. Perelman, Finite extinction time for the solutions of the Ricci flow on certain three-manifolds, arXiv: math. DG/0307245 (2003)

  36. C. Rourke, B. Sanderson, Introduction to Piecewise Linear Topology (Springer, Berlin, 1969)

    MATH  Google Scholar 

  37. S. Smale, On the structure of \(5\)-manifolds. Ann. Math. 75(1), 38–46 (1962)

    Article  MathSciNet  Google Scholar 

  38. F. Spaggiari, Regular genus and products of spheres. J. Korean Math. Soc. 47(5), 925–934 (2010)

    Article  MathSciNet  Google Scholar 

  39. E. Swartz, Face enumeration- from spheres to manifolds. J. Eur. Math. Soc. 11, 449–485 (2009)

    Article  MathSciNet  Google Scholar 

  40. V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Math. 18 ed. by H. Bauer, J.L. Kazdan, E. Zehnder. (Walter de Gruyter & Co., Berlin, Germany, 1994)

  41. A. Vince, n-Graphs. Discrete Math. 72, 367–380 (1988)

    Article  MathSciNet  Google Scholar 

  42. G.M. Ziegler, Lectures on Polytopes Graduate Texts in Math (Springer, Berlin, 1995)

    Book  Google Scholar 

Download references

Acknowledgements

This work was performed under the auspices of the “National Group for Algebraic and Geometric Structures, and their Applications” of the GNSAGA-INDAM within the CNR (National Research Council) of Italy and partially supported by the MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) of Italy within the project “Strutture Geometriche, Combinatoria e loro Applicazioni”. We should like to thank the Managing Editor of the journal, Professor András I. Stipsicz and three anonymous referees for their useful comments and suggestions improving the original version of the paper.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100, Modena, Italy

    Alberto Cavicchioli & Fulvia Spaggiari

Authors
  1. Alberto Cavicchioli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fulvia Spaggiari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Cavicchioli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cavicchioli, A., Spaggiari, F. On reduced complexity of closed piecewise linear 5-manifolds. Period Math Hung 83, 144–158 (2021). https://doi.org/10.1007/s10998-020-00375-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-020-00375-6

Keywords

Mathematics Subject Classification

Search

Navigation