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The rack space. (English) Zbl 1123.55006

Authors’ abstract: The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle. We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for \( \pi_2\) of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

MSC:

55Q40 Homotopy groups of spheres
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R20 Characteristic classes and numbers in differential topology
57R40 Embeddings in differential topology
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