Link maps and the geometry of their invariants
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 \(\pi_{p+q}(S^{m-1})\).
scientific article; zbMATH DE number 4129656
scientific article