Immersions and embeddings of orientable manifolds up to unoriented cobordism
type: Item
instance of: scholarly article
title: Immersions and embeddings of orientable manifolds up to unoriented cobordism
zbMATH Open document ID: 0883.57029
author: Isao Takata
published in: Hiroshima Mathematical Journal
publication date: 1998-03-09T00:00:00Z
review text: Let \(\alpha(n)\) denote the number of ones in the binary expansion of \(n\). \textit{R. L. W. Brown} [Can. J. Math. 23, 1102-1115 (1971; Zbl 0216.20104)] has proved that every differentiable closed \(n\)-manifold \(M^n\) is cobordant to a manifold that immerses in \(R^{2n-\alpha (n)}\) and embeds into \(R^{2n- \alpha(n)+1}\). If some additional conditions are placed on \(M^n\), then it is possible to lower the dimension to \(2n-\alpha (n)\), respectively, \(2n-\alpha (n) +1\). Some such conditions can be already found in Brown's cited paper. The author now places the orientability of \(M^n\) as the additional condition in Brown's theorem. Writing \(n\) in the form \(n= (2m+1) 2^{\nu(n)}\), let \(\beta(n) =2n- \alpha(n)- \min \{\alpha(n), \nu(n)\}\). Then as the main result one could consider the following statement: Any orientable manifold \(M^n\) immerses (embeds) up to unoriented cobordism in \(R^{\beta(n)}\) \((R^{\beta (n)+1})\). Also, depending on certain properties of \(n\), \(\beta(n)\) and \(\nu(n)\) it is shown that in such cases \(M^n\) immerses (embeds) up to unoriented cobordism in \(R^{\beta (n)-1}\) \((R^{\beta (n)})\).
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 57R75
zbMATH DE Number: 1021653
zbMATH Keywords: orientable manifold
zbMATH Keywords: unoriented cobordism
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206127142
OpenAlex ID: W1480563786
Wikidata QID: Q128871112
DOI: 10.32917/HMJ/1206127142
description: scientific article; zbMATH DE number 1021653