Obstructions to approximating maps of \(n\)-manifolds into \(\mathbb{R}^{2n}\) by embeddings
type: Item
instance of: scholarly article
title: Obstructions to approximating maps of \(n\)-manifolds into \(\mathbb{R}^{2n}\) by embeddings
zbMATH Open document ID: 1003.57036
DOI: 10.1016/S0166-8641(01)00164-X
published in: Topology and its Applications
publication date: 2002-09-17T00:00:00Z
review text: The first result, which is proved in one paragraph, is that if \(n\geq 3\), then a general position map \(f\) from a closed \(n\)-manifold \(K\) into \({\mathbb R}^{2n-1}\) is the projection of an embedding into \({\mathbb R}^{2n}\) if and only if the composite \(K\to{\mathbb R}^{2n-1} \hookrightarrow{\mathbb R}^{2n}\) is approximable by embeddings. The main result involves the case \(n=2\). If \(K\) is a closed 2-manifold and \(f:K\to {\mathbb R}^3\) is a general position map, an obstruction \(\beta(f)\) to approximability of the composite \(K\to{\mathbb R}^3\hookrightarrow\mathbb{R}^4\) by embeddings is defined, using resolvability of triple points. Certain quadratic forms \(q_T\) associated to the intersection form on \(H^1(K;{\mathbb Z}_2)\) are defined, and it is proved that \(\beta(f)\) equals the product of the Arf invariants of the forms \(q_T\).
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57M20
zbMATH DE Number: 1801745
zbMATH Keywords: embeddings
zbMATH Keywords: Arf invariant
author: Petr Akhmet'ev
author: Dušan Repovš
author: A. B. Skopenkov
reviewed by: Donald M. Davis
MaRDI profile type: Publication
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full work available at URL: s0166-8641(01)00164-x
OpenAlex ID: W2024962218
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description: scientific article; zbMATH DE number 1801745