Regular genus and gem-complexity of some mapping tori
Biplab Basak

TL;DR
This paper constructs crystallizations of mapping tori for certain PL-manifolds, providing sharp bounds for regular genus and gem-complexity, and disproving a conjecture about regular genus characterization.
Contribution
It introduces a method to construct crystallizations of mapping tori, establishing upper bounds for regular genus and gem-complexity, and disproving a conjecture related to regular genus characterization.
Findings
Sharp bounds for regular genus of mapping tori.
Construction of crystallizations for specific PL-manifolds.
Disproof of a conjecture linking regular genus six to topological products.
Abstract
In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms for a certain class of PL-manifolds . 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 , where is , , , , , \mathbb{S}^{\hspace{.2mm}2} \mbox{\times \hspace{-2.6mm}_{-}} \, \mathbb{S}^{\hspace{.1mm}1} or . In particular, for 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 with regular genus…
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