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−6 and G(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.

Lower bounds for regular genus and gem-complexity of PL 4-manifolds / Basak, B.; Casali, Maria Rita. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - STAMPA. - 29:4(2017), pp. 761-773. [10.1515/forum-2015-0080]

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

CASALI, Maria Rita
2017

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−6 and G(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.
2017
11-ago-2016
29
4
761
773
Lower bounds for regular genus and gem-complexity of PL 4-manifolds / Basak, B.; Casali, Maria Rita. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - STAMPA. - 29:4(2017), pp. 761-773. [10.1515/forum-2015-0080]
Basak, B.; Casali, Maria Rita
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1108461
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