On the Haefliger-Hirsch-Wu invariants for embeddings and immersions
type: Item
instance of: scholarly article
title: On the Haefliger-Hirsch-Wu invariants for embeddings and immersions
zbMATH Open document ID: 1012.57035
published in: Commentarii Mathematici Helvetici
publication date: 2002-06-17T00:00:00Z
review text: This work is in the PL category and treats the isotopy classes of embeddings of closed PL manifolds into Euclidean spaces and regular concordance classes of immersions of polyhedra into Euclidean spaces. We shall present two sample results which in the reviewer's opinion give an idea about the content of this extensive study. Already at the beginning of the investigation of the possibility of embedding an \(n\)-dimensional polyhedron \(K\) into the \(2n\)-dimensional Euclidean space \(\mathbb R^{2n}\), E. R. van Kampen has used the so-called simplicial deleted product \(\widetilde T \subset T \times T\) consisting of products \(\sigma \times \tau\) of disjoint simplices, \(\sigma \cap \tau = \emptyset\), of a triangulation \(T\) of \(K\). He constructed an obstruction to embeddability of an \(n\)-polyhedron into \(\mathbb R^{2n}\) which was developed further. From that time on, \(\widetilde T\) and the deleted product \(\widetilde K = K \times K \smallsetminus \Delta\) (where \(\Delta\) denotes the diagonal of \(K \times K\)) are significantly used in the theory of embeddings and immersions in the PL and the differentiable category. To each embedding \(f: K \rightarrow \mathbb R^m\) one assigns a map \(\widetilde f : \widetilde K \rightarrow S^{m-1}\) by \(\widetilde f(x,y) = (f(x) - f(y))/ |f(x) -f(y) |\). \(\mathbb Z_2\) acts on \(\widetilde K\) and \(S^{m-1}\) freely by permuting factors of \(\widetilde K\) and by antipodes, respectively. Since \(\widetilde f\) is a \(\mathbb Z_2\)-equivariant map it is a representative of the equivariant homotopy class, denoted \(\alpha (f)\), of equivariant maps \(\widetilde K: \rightarrow S^{m-1}\). Denote by \(\pi_{\text{eq}}^{m-1}(\widetilde K)\) the set of equivariant homotopy classes of \(\widetilde K\) into \(S^{m-1}\) . Since \(\alpha (f)\) is clearly an isotopy invariant of \(f\), one obtains a function \(\alpha^m(K) : \text{Emb}^m(K) \rightarrow \pi_{\text{eq}}^{m-1}(\widetilde K)\), which in this paper is called Haefl…
reviewed by: Ivan Ivanšić
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 57Q37
Mathematics Subject Classification ID: 57Q60
zbMATH DE Number: 1756819
zbMATH Keywords: embedding
zbMATH Keywords: immersion
zbMATH Keywords: deleted product
zbMATH Keywords: metastable dimension range
zbMATH Keywords: isotopy
author: A. B. Skopenkov
MaRDI profile type: Publication
full work available at URL: s00014-002-8332-4
OpenAlex ID: W1986613059
DOI: 10.1007/S00014-002-8332-4
description: scientific article; zbMATH DE number 1756819