Link maps and the geometry of their invariants
type: Item
instance of: scholarly article
title: Link maps and the geometry of their invariants
zbMATH Open document ID: 0689.57016
DOI: 10.1007/BF01258596
author: Ulrich Koschorke
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155322
review text: This paper investigates several invariants of link homotopy. A link map consists of two maps \(f_ 1: S^ p\to S^ m\), \(f_ 2: S^ q\to S^ m\) whose images are disjoint; a link homotopy is a one-parameter continuous family of such maps. \(LM^ m_{p,q}\) denotes the set of link homotopy classes. It is shown that connected sum makes \(LM^ m_{p,q}\) an abelian semi-group if p or \(q\leq m-3\) or if p and \(q\leq m-2\). The existence of inverses is discussed. The author defines two invariants \(\alpha\) : \(LM^ m_{p,q}\to \pi^ s_ n\), the stable n-stem, where \(n=p+q+1-m\), and \(\beta\) : \(LM^ m_{p,q}\to \Omega_{2n- p}(P^{\infty};(p-n)\lambda)\) where \(\Omega_ i(X+\xi)\) is the normal bordism classes of (f,\(\Phi)\), f: \(M\to X\) and \(\Phi\) a trivialization of \(f^*\xi \oplus \tau_ M\) (M is a closed i-manifold and \(\xi\) a vector bundle over X) and \(\lambda\) the canonical line bundle. \(\alpha\) is a classical link homotopy invariant, while \(\beta\) measures the self- intersections of \(f_ 2(S^ q)\) and their linking with \(f_ 1(S^ p)\). One of the main result is that these invariants are related, via a ``double-point Hopf invariant'' \(h_{n,p}: \pi^ s_ n\to \Omega_{2n-p}(P^{\infty};(p-n)\lambda)\), to a usual Hopf invariant. Two constructions of link maps define homomorphisms \(e_*: \pi_ p(S^{m-q-1})\to LN^ m_{p,q}\), K: \(\pi_ n(SO_{p+q-2n}))\to LM^ m_{p,q}\). If \(q<m-2\), Im \(e_*\) is exactly those elements represented by a link map with \(f_ 2\) an imbedding. In a later paper, the author defines a more delicate invariant \({\tilde \beta}\) whose kernel is exactly Im \(e_*\) in a ``metastable'' range of dimensions. Both \(e_*\) and K are used to demonstrate the realizability of certain elements of \(\pi^ s_ n\) by \(\alpha\). Finally the author discusses a double filtration of \(\pi^ s_ n\) defined by \(\pi_ n^{p,q}=\alpha (LM^ m_{p,q})\), and, in particular its relation to the stable suspension of the groups \(\pi_ p(S^{p-n})\) and \(\pi_{p+q}(S^{m-1})\).
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 55Q25
zbMATH DE Number: 4129656
zbMATH Keywords: K-invariant
zbMATH Keywords: stable homotopy groups of spheres
zbMATH Keywords: invariants of link homotopy
zbMATH Keywords: link homotopy classes
zbMATH Keywords: normal bordism classes
zbMATH Keywords: self- intersections
zbMATH Keywords: double-point Hopf invariant
reviewed by: Jerome Levine
MaRDI profile type: Publication
cites work: On the non-existence of elements of Hopf invariant one
cites work: Étude homotopique des espaces de plongements
cites work: Spheres May Link Homotopically in 4-Space
cites work: Immersions of Manifolds
cites work: ON THE ITERATED SUSPENSION
cites work: An interpretation of G. Whitehead's generalization of H. Hopf's invariant
cites work: Vector fields and other vector bundle morphisms - a singularity approach
cites work: Q3700455
cites work: On link maps and their homotopy classification
cites work: Higher-dimensional link operations and stable homotopy
cites work: Geometric interpretations of the generalized Hopf invariant.
cites work: Self-intersections and higher Hopf invariants
cites work: Q3696051
cites work: Link groups
cites work: On Dehn's lemma and the asphericity of knots
cites work: Bordism and Geometric Dimension.
cites work: Homotopy links
cites work: The classification of immersions of spheres in Euclidean spaces
cites work: Compositional Methods in Homotopy Groups of Spheres. (AM-49)
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description: scientific article; zbMATH DE number 4129656