Manifolds and discrete structures
type: Item
instance of: scholarly article
title: Manifolds and discrete structures
zbMATH Open document ID: 0536.57017
published in: Kodai Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: The starting point of this paper is the connection between smooth structures on a manifold M, the algebra of smooth functions on M and the behaviour of smooth functions on a dense subset of M. It is shown that if \((a_ n)\) and \((b_ n)\) are dense sequences in M with \(a_ m\neq a_ n\) and \(b_ m\neq b_ n\) for \(m\neq n\) then there is a diffeomorphism \(\phi\) : \(M\to M\) with \(\phi(\{a_ n| n=1,...,\infty \})=\{b_ n| n=1,...,\infty \}\). A discrete structure on M is an equivalence class of dense sequences, where two sequences \((a_ n)\) and \((b_ n)\) as above are equivalent if there is a diffeomorphism \(\phi\) of M and a positive integer N such that for each \(n\geq N\), \(\phi(a_ n)=b_ n\). There is a natural relationship between the set of discrete structures on M and a certain coset space of the symmetric group on the positive integers. A condition under which two sequences give rise to the same discrete structure is given.
Mathematics Subject Classification ID: 57R99
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 58C05
zbMATH DE Number: 3852042
zbMATH Keywords: discrete structures on manifolds
zbMATH Keywords: equivalence class of dense sequences
author: Kôji Shiga
reviewed by: David B. Gauld
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138036717
OpenAlex ID: W2022449971
cites work: Q5331572
DOI: 10.2996/KMJ/1138036717
description: scientific article; zbMATH DE number 3852042