Genus-minimal crystallizations of PL 4-manifolds

TL;DR

This paper characterizes when a PL 4-manifold's regular genus attains a minimal value using weak semi-simple crystallizations and explores its additivity under connected sums, linking to the 4D Smooth Poincaré Conjecture.

Contribution

It introduces weak semi-simple crystallizations for PL 4-manifolds and establishes a criterion for minimal regular genus, also proving additivity under connected sum for these manifolds.

Findings

Regular genus equals the minimal value if and only if the manifold admits a weak semi-simple crystallization.

Regular genus is additive under connected sum for manifolds with weak semi-simple crystallizations.

The property relates to the 4-dimensional Smooth Poincaré Conjecture.

Abstract

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL $4$-manifold $M$, it is known that its regular genus $\mathcal G(M)$ is at least $2 \chi (M) + 5m -4$, where $m$ is the rank of the fundamental group of $M$. In this article, we introduce the concept of "weak semi-simple crystallization" for every closed connected PL $4$-manifold $M$, and prove that $\mathcal G(M)= 2 \chi (M) + 5m -4$ if and only if $M$ admits a weak semi-simple crystallization. We then show that the PL invariant regular genus is additive under the connected sum within the class of all PL 4-manifolds admitting a weak semi-simple crystallization. Also, we note that this property is related to the 4-dimensional Smooth Poincar\'e Conjecture.