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

Biplab Basak and Maria Rita Casali
From the journal Forum Mathematicum

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)3χ(M)+10m-6and𝒢(M)2χ(M)+5m-4,

where rk(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.


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.

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

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