Codimension One Foliation and the Prime Spectrum of a Ring

Let F be a transversally oriented codimension-one foliation of class Cr, r ≥ 0, on a closed manifold M. A leaf class of a leaf F is the union of all leaves having the same closure as F . Let X be the leaf classes space and X0 be the union of all open subsets of X homeomorphic to R or S1. In [2, Theo...

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Bibliographic Details
Published in:European journal of pure and applied mathematics 2024-01, Vol.17 (1), p.356-361
Main Author: Alharbi, Badr
Format: Article
Language:English
Online Access:Get full text
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Summary:Let F be a transversally oriented codimension-one foliation of class Cr, r ≥ 0, on a closed manifold M. A leaf class of a leaf F is the union of all leaves having the same closure as F . Let X be the leaf classes space and X0 be the union of all open subsets of X homeomorphic to R or S1. In [2, Theorem 3.15] it is shown that if a codimension one foliation has a finite height, then the singular part of the space of leaf classes is homeomorphic to the prime spectrum (or simply the spectrum) of unitary commutative ring. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below.
ISSN:1307-5543
1307-5543
DOI:10.29020/nybg.ejpam.v17i1.4803