On isotopy classification of embeddings of \(n\)-manifolds in \(\mathbb{RP}^{2n}\)
type: Item
instance of: scholarly article
title: On isotopy classification of embeddings of \(n\)-manifolds in \(\mathbb{RP}^{2n}\)
zbMATH Open document ID: 0824.57021
DOI: 10.1016/0166-8641(93)90123-U
author: Ping Zhang
published in: Topology and its Applications
publication date: 1994-03-01T00:00:00Z
review text: The isotopy group of embeddings, in the metastable range, of an orientable closed manifold \(M\) in the real projective space of even dimension, is computed in terms of the homotopy groups of \(M\). The computation is based on the approach to embedding problems in normal bordism framework, which converts the calculations into those of singular homology groups of a certain elaborate space pair with twisted integers as coefficients. It forms part of a project which applies the normal bordism method to the existence and isotopy classification of differential embeddings of manifolds in manifolds in the metastable range. An example of the type of results obtained is the following Theorem: Suppose that \(n \geq 4\). Let \(M^ n\) be a closed connected manifold and let \(f : M^ n \to \mathbb{RP}^{2n}\) be a map which induces an epimorphism \(f_ * : \pi_ 1 (M) \to \pi_ 1 (\mathbb{RP}^ 2n) = \mathbb Z_ 2\). Assume that there is no element in \(\ker f_ *\) which is represented by orientation reversing loops (e.g. when \(M\) is orientable). Then, \[ [M^ n \subset \mathbb{RP}^{2m}]_ f = \begin{cases} \bigl( \ker f_ */ [\ker f_ *, \ker f_ *] \bigr) \otimes \mathbb Z_ 2 \quad & \text{if}\;n \;\text{is even}, \\ \ker\;f_ * / [\ker\;f_ *,\ker\;f_ *]G, \quad & \text{if}\;n\;\text{is odd}, \end{cases} \] where \(G\) stands for the subgroup of \(\ker\;f_ *\) generated by the squares of elements which are not in \(\ker\;f_ *\), and where \([M \subset N]_ f\) denotes \(\pi_ 1 (N^ M, E,f)\) for \(E\) the space of differential embeddings and \(f : M \to N\).
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 57R75
zbMATH DE Number: 500680
zbMATH Keywords: homology with local coefficients
zbMATH Keywords: isotopy group of embeddings
zbMATH Keywords: metastable range
zbMATH Keywords: normal bordism
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 500680