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Regular genus and gem-complexity of some mapping tori

  • Biplab BasakEmail author
Original Paper
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Abstract

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.

Keywords

PL-manifolds Mapping torus Crystallizations Regular genus Gem-complexity 

Mathematics Subject Classification

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

Notes

Acknowledgements

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).

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

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

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