The rack space

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2007-02, Vol.359 (2), p.701-740
Main Authors: Fenn, Roger, Rourke, Colin, Sanderson, Brian
Format: Article
Language:English
Subjects:
Algebraic topology
Cubes
Diagrams
Exact sciences and technology
Homomorphisms
Isotopic labeling
Labor union representation
Manifolds and cell complexes
Mathematical manifolds
Mathematical theorems
Mathematics
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Vertices
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-06-03912-2