A generalized Conner--Floyd conjecture and the immersion problem for low 2-torsion lens spaces
type: Item
instance of: scholarly article
title: A generalized Conner--Floyd conjecture and the immersion problem for low 2-torsion lens spaces
zbMATH Open document ID: 1022.57017
DOI: 10.1016/S0040-9383(02)00084-8
author: Jesús González
published in: Topology
publication date: 2003-04-27T00:00:00Z
review text: This paper gives new non-immersion results for \(2^k\)-torsion lens spaces, for \(k\) small, into Euclidean space. Let \(\alpha(d)\) denote the number of ones in the binary expansion of \(d\). The main result of the paper is the following theorem: ``For \(1\leq k\leq\alpha(d)\) the lens space \(L(2^k)^{2(d+\alpha(d)- K)+ 1}\) cannot be immersed into a Euclidean space of dimension \(4d-2\alpha(d)\) if either one of the following conditions hold: (i) \(\alpha(d)- k\leq 6\), (ii) \(2^{v(d)}> \alpha(d)-k\), where \(v(d)\) is the highest power of 2 dividing \(d\).'' Results for \(k\geq \alpha(d)\) were obtained in [\textit{J. Gonzalez}, J. Lond. Math. Soc. 63, No. 1, 247-256 (2001; Zbl 1013.57017)]. Furthermore, the above result for \(k=1\) corresponds to Davis' strong non-immersion theorem for real projective spaces. The paper uses the Brown-Peterson spectrum. The non-immersion is obtained by showing that the Euler class of a certain bundle \(n\) is non-trivial. Such obstructions are BP\(_*\)-multiples of the toral class \(\gamma_n\). So, the annihilator ideal \(I_n= \text{Ann}(\gamma_n)\) plays an important role. The paper contains a good description of the known results and, compared with the techniques used in related works, the simple theoretical approach (based on BP-techniques) suggests one should expect a similar type of solution in other cases.
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 55S35
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 1900978
zbMATH Keywords: Lens spaces
zbMATH Keywords: immersions of manifolds
zbMATH Keywords: Brown-Peterson homology
zbMATH Keywords: Adam's spectral sequence
zbMATH Keywords: Conner-Floyd conjecture
Wikidata QID: Q122889062
MaRDI profile type: Publication
full work available at URL: s0040-9383(02)00084-8
OpenAlex ID: W2070641514
description: scientific article; zbMATH DE number 1900978