Some new immersions and nonimmersions of \(2^r\)-torsion lens spaces
type: Item
instance of: scholarly article
title: Some new immersions and nonimmersions of \(2^r\)-torsion lens spaces
zbMATH Open document ID: 1042.57015
author: Thomas A. Shimkus
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2004-03-30T00:00:00Z
review text: Let \(L^{2n+1}(2^r)\) be the \((2n+1)\)-dimensional \(2^r\)-torsion lens space, for positive integers \(n\) and \(r\). This work deals with the immersion of these spaces in Euclidean spaces. One relevant number associated to the immersion problem of a manifold of dimension \(n\) in Euclidean spaces is \(\alpha(n)\), which is the number of \(1's\) in the binary expansion of \(n\). The author obtains some results for small values of \(\alpha(n)\), namely for \(\alpha(n)=2\) and \(\alpha(n)=3\), where most of them are optimal. We exemplify with two such results: Theorem (1.1) If \(n\) is odd and \(\alpha(n)=2\), then \(L^{2n+1}(4)\) immerses in \(\mathbb{R}^{4n-2\alpha(n)+1}\) and, for \(r\geq 3\), \(L^{2n+1}(2^r)\) does not immerse in \(\mathbb{R}^{4n-2\alpha(n)+1}\). Theorem(1.4) If \(n\equiv 2 \bmod 4\), \(\alpha(n)=3\) and \(r\geq 2\), then \(L^{2n+1}(2^r)\) does not immerse in \(\mathbb{R}^{4n-2\alpha(n)}\). Modified Postnikov towers have already been used for this type of problem with success. In order to prove the results he uses a modified version of the Postnikov tower of the map \(\widetilde{BU}(2n-3) \to BU\), where this map is the induced fibration from the \(BO(2n-3) \to BO\) map by the retraction \(r:BU \to BO\). This modified Postnikov tower is obtained using Adams spectral sequence (ASS) of the stunted real projective space. One should decide if a particular map lifts or not to a certain stage of this modified Postnikov tower. To solve this problem the author uses complex connective \(K\)-theory.
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 55S35
zbMATH DE Number: 2063649
zbMATH Keywords: Postnikov tower
zbMATH Keywords: lens space
zbMATH Keywords: immersion
zbMATH Keywords: obstruction theory
zbMATH Keywords: Adams spectral sequence
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2063649