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

TL;DR

This paper establishes new lower bounds for the gem-complexity and regular genus of closed connected PL 4-manifolds, improving previous estimates and introducing semi-simple crystallizations that attain these bounds.

Contribution

It proves lower bounds for PL 4-manifolds' invariants, introduces semi-simple crystallizations, and shows additivity of these invariants under connected sum.

Findings

Lower bounds for gem-complexity and regular genus in terms of Euler characteristic and fundamental group rank.

Semi-simple crystallizations attain the established lower bounds.

Additivity of invariants under connected sum for semi-simple crystallizations.

Abstract

Within crystallization theory, two interesting PL invariants for $d$-manifolds have been introduced and studied, namely {\it gem-complexity} and {\it regular genus}. In the present paper we prove that, for any closed connected PL $4$-manifold $M$, its gem-complexity $\mathit{k}(M)$ and its regular genus $ \mathcal G(M)$ satisfy: $$\mathit{k}(M) \ \geq \ 3 \chi (M) + 10m -6 \ \ \ \text{and} \ \ \ \mathcal G(M) \ \geq \ 2 \chi (M) + 5m -4,$$ 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 {\it 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 catalogues, and their connected sums.