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Parallelizability of algebraic knots and canonical framings. (English) Zbl 0583.57012

Let f: (\({\mathbb{C}}^ n,0)\to ({\mathbb{C}},0)\) be a holomorphic germ with isolated singularity 0. Denote the disc in \({\mathbb{C}}^ n\) centered at 0 with radius \(\epsilon\) by \(D_{\epsilon}\) and \(S_{\epsilon}=\partial D_{\epsilon}\). For small \(\epsilon\), \(M=f^{-1}(0)\cap S_{\epsilon}\) is a manifold and is called the algebraic knot of f.
In this paper, for \(n=3\) and 5, a canonical framing on M is introduced by using immersion theory, which generalizes the canonical framing of J. A. Seade [Topology 21, 1-8 (1981; Zbl 0477.57021)] for \(n=3\). Further, it is determined that the element in the stable homotopy group of spheres \(\pi^ s_{2n-3}\) given by the canonical framing is \((\mu +1)\nu\), where \(\mu\) is the Milnor number of M and \(\nu\) is the J-image of a suitable generator of \(\pi_{2n-3}(S0).\)
Let \(\chi^*(M)\) be the semi-characteristic number of M. It is proved that \(\mu +1\equiv \chi^*(M) mod 2.\) This shows that the mod 2 Milnor number depends only upon the homotopy type of M, also yields a parallelizability criterion for algebraic knots: If \(n\neq 2,3,5\), then M is parallelizable iff \(\mu\) is odd (if \(n=2,3,5\), M is always parallelizable).

MSC:

57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
32Sxx Complex singularities
55Q45 Stable homotopy of spheres
55Q50 \(J\)-morphism
57R42 Immersions in differential topology

Citations:

Zbl 0477.57021
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