Regular genus and gem-complexity of some mapping tori

Abstract

In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms $f:M\to M$ for a certain class of PL-manifolds $M$. These yield upper bounds for gem-complexity and regular genus of a large class of PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL) homeomorphisms $f:M\to M$, where $M$ is ${\mathbb{RP}}^2$, ${\mathbb{RP}}^2\#{\mathbb{RP}}^2$, ${\mathbb{S}}^1\times {\mathbb{S}}^1$, ${\mathbb{RP}}^3$, ${\mathbb{S}}^2\times {\mathbb{S}}^1$, $\mathbb{S}^{\hspace{.2mm}2} \mbox{$\times \hspace{-2.6mm}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$ or ${\mathbb{S}}^d$. In particular, for $M={\mathbb{S}}^{d-1}\times {\mathbb{S}}^1$ or $\mathbb{S}^{\hspace{.2mm}d-1} \mbox{$\times\hspace{-2.6mm}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$, our construction gives a crystallization of a mapping torus of a (PL) homeomorphism $f:M\to M$ with regular genus $d^2-d$. As a consequence, we prove the existence of an orientable mapping torus of a (PL) homeomorphism $f:({\mathbb{S}}^2\times {\mathbb{S}}^1)\to ({\mathbb{S}}^2\times {\mathbb{S}}^1)$ with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product ${\mathbb{RP}}^3\times {\mathbb{S}}^1$ among closed connected prime orientable PL $4$-manifolds.

16 pages, 8 figures; Published online on 24 January 2019 in the journal RACSAM. DOI: https://doi.org/10.1007/s13398-019-00634-3

References in corpus (4)

• Topology in colored tensor models via crystallization theory
• G-degree for singular manifolds
• Lower bounds for regular genus and gem-complexity of PL 4-manifolds
• Combinatorial properties of the G-degree

Cited by in corpus (3)

• 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 ${\Delta }^{n-1}\times I$