Genus-minimal crystallizations of PL 4-manifolds

Abstract

For $d\ge 2$, 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 Poincaré Conjecture.

10 pages, no figure. Minor correction (cf. Remark 14) in Lemma 7. arXiv admin note: text overlap with arXiv:1504.00771

References in corpus (2)

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

Cited by in corpus (5)

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• Minimal crystallizations of 3-manifolds with boundary
• Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
• Crystallizations of small covers over the $n$-simplex ${\Delta }^n$ and the prism ${\Delta }^{n-1}\times I$
• Handle decompositions for a class of closed orientable PL 4-manifolds