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

Abstract

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 ) {k(M)} and its regular genus 𝒢 ⁢ ( M ) {\mathcal{G}(M)} satisfy k ⁢ ( M ) ≥ 3 ⁢ χ ⁢ ( M ) + 10 ⁢ m - 6 and 𝒢 ⁢ ( M ) ≥ 2 ⁢ χ ⁢ ( M ) + 5 ⁢ m - 4 , k(M)\geq 3\chi(M)+10m-6\quad\text{and}\quad\mathcal{G}(M)\geq 2\chi(M)+5m-4, where r ⁢ k ⁢ ( π 1 ⁢ ( M ) ) = m {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.