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

Basak, Biplab and Casali, Maria Rita (2017) Lower bounds for regular genus and gem-complexity of PL 4-manifolds. In: FORUM MATHEMATICUM, 29 (4). pp. 761-773.

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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 chi(M) + 10m - 6 and G(M) >= 2 chi(M) + 5m - 4, where rk(pi(1) (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.

Item Type:	Journal Article
Publication:	FORUM MATHEMATICUM
Additional Information:	Copy right for this article belongs to the WALTER DE GRUYTER GMBH, GENTHINER STRASSE 13, D-10785 BERLIN, GERMANY
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	05 Aug 2017 09:32
Last Modified:	05 Aug 2017 09:32
URI:	http://eprints.iisc.ac.in/id/eprint/57577

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