Regular genus and gem-complexity of some mapping tori

  • Biplab BasakEmail author
Original Paper


In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms \(f:M \rightarrow M\) for a certain class of PL-manifolds M. These yield upper bounds for gem-complexity and regular genus of a large class of PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL) homeomorphisms \(f:M \rightarrow M\), where M is \(\mathbb {RP}^2\), \(\mathbb {RP}^2\#\mathbb {RP}^2\), \(\mathbb {S}^1\times \mathbb {S}^1\), \(\mathbb {RP}^3\), \(\mathbb {S}^{2} \times \mathbb {S}^1\), \(\mathbb {S}^{2} \times _{-} \, \mathbb {S}^{1}\) or \(\mathbb {S}^d\). In particular, for \(M=\mathbb {S}^{d-1} \times \mathbb {S}^1\) or \(\mathbb {S}^{d-1} \times _{-} \, \mathbb {S}^{1}\), our construction gives a crystallization of a mapping torus of a (PL) homeomorphism \(f:M \rightarrow M\) with regular genus \(d^2-d\). As a consequence, we prove the existence of an orientable mapping torus of a (PL) homeomorphism \(f:(\mathbb {S}^{2} \times \mathbb {S}^1)\rightarrow (\mathbb {S}^{2} \times \mathbb {S}^1)\) with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product \(\mathbb {RP}^3 \times \mathbb {S}^1\) among closed connected prime orientable PL 4-manifolds.


PL-manifolds Mapping torus Crystallizations Regular genus Gem-complexity 

Mathematics Subject Classification

Primary 57Q15 Secondary 05C15 57N10 57N13 57N15 57Q05 55R10 



The author would like to thank Basudeb Datta for useful suggestions. The author is also thankful to the anonymous referees for many useful comments and suggestions. In particular, Conjecture 18 is due to one of them. The author is supported by DST INSPIRE Research Grant (DST/INSPIRE/04/2017/002471).


  1. 1.
    Bandieri, P., Casali, M.R., Gagliardi, C.: Representing manifolds by crystallization theory: foundations, improvements and related results. Atti Sem. Mat. Fis. Univ. Modena 49(suppl.), 283–337 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bandieri, P., Cristofori, P., Gagliardi, C.: Nonorientable 3-manifolds admitting colored triangulations with at most 30 tetrahedra. J. Knot Theory Ramif. 18, 381–395 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Basak, B., Casali, M.R.: Lower bounds for regular genus and gem-complexity of PL 4-manifolds. Forum Math. 29(4), 761–773 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basak, B., Datta, B.: Minimal crystallizations of 3-manifolds. Electron. J Combin. 21(1), P1.61, 1–25 (2014)Google Scholar
  5. 5.
    Björner, A.: Posets, regular CW complexes and Bruhat order. Eur. J. Comb. 5, 7–16 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Casali, M.R.: Classification of nonorientable 3-manifolds admitting decompositions into \(\le \) 26 coloured tetrahedra. Acta Appl. Math. 54, 75–97 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casali, M.R., Cristofori, P.: A catalogue of orientable 3-manifolds triangulated by 30 colored tetrahedra. J. Knot Theory Ramif. 17, 579–599 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casali, M.R., Cristofori, P.: Cataloguing PL 4-manifolds by gem-complexity. Electron. J. Combin. 22(4), #P4.25, 1–25 (2015)Google Scholar
  10. 10.
    Casali, M.R., Cristofori, P., Dartois, S., Grasselli, L.: Topology in colored tensor models via crystallization theory. J. Geom. Phys. 129, 142–167 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Casali, M.R., Cristofori, P., Grasselli, L.: \(G\)-degree for singular manifolds. RACSAM 112(3), 693–704 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Casali, M.R., Grasselli, L.: Combinatorial properties of \(G\)-degree. Rev. Mat. Complut. (2018).
  13. 13.
    Cavicchioli, A.: On the genus of smooth 4-manifolds. Trans. Am. Math. Soc. 31, 203–214 (1999)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Cavicchioli, A., Meschiari, M.: On classification of 4-manifolds according to genus. Cahiers Topologie Geom. Differentielle Categoriques 34, 37–56 (1993)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cavicchioli, A., Spaggiari, F.: On the genus of real projective spaces. Arch. Math. 89, 570–576 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cristofori, P.: On the genus of \(\mathbb{S}^m \times \mathbb{S}^n\). J. Korean Math. Soc. 41(3), 407–421 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cristofori, P., Mulazzani, M.: Compact 3-manifolds via 4-colored graphs. RACSAM 110, 395–416 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ferri, M., Gagliardi, C.: On the genus of \(4\)-dimensional products of manifolds. Geom. Dedicata 13, 331–345 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ferri, M., Gagliardi, C.: The only genus zero \(n\)-manifold is \({S}^n\). Proc. Am. Math. Soc. 85, 638–642 (1982)zbMATHGoogle Scholar
  20. 20.
    Ferri, M., Gagliardi, C., Grasselli, L.: A graph-theoretic representation of PL-manifolds–a survey on crystallizations. Acquationes Math. 31, 121–141 (1986)CrossRefzbMATHGoogle Scholar
  21. 21.
    Gagliardi, C.: How to deduce the fundamental group of a closed \(n\)-manifold from a contracted triangulation. J. Comb. Inf. Syst. Sci. 4, 237–252 (1979)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gagliardi, C.: Extending the concept of genus to dimension \(n\). Proc. Am. Math. Soc. 81, 473–481 (1981)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Gagliardi, C., Grasselli, L.: Representing products of polyhedra by products of edge-colored graphs. J. Graph Theory 17, 549–579 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gagliardi, C., Volzone, G.: Handles in graphs and sphere bundles over \(\mathbb{S}^1\). Eur. J. Comb. 8, 151–158 (1987)CrossRefzbMATHGoogle Scholar
  25. 25.
    Pezzana, M.: Sulla struttura topologica delle varietà compatte. Atti Sem. Mat. Fis. Univ. Modena 23, 269–277 (1974)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Spaggiari, F.: On the genus of \({\mathbb{RP}}^3 \times {S}^1\). Collect. Math. 50(3), 229–241 (1999)MathSciNetGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiHauz Khaz, New DelhiIndia

Personalised recommendations