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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The rack space


Authors: Roger Fenn, Colin Rourke and Brian Sanderson
Journal: Trans. Amer. Math. Soc. 359 (2007), 701-740
MSC (2000): Primary 55Q40, 57M25; Secondary 57Q45, 57R15, 57R20, 57R40
Published electronically: August 24, 2006
MathSciNet review: 2255194
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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.


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Additional Information

Roger Fenn
Affiliation: Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom
Email: R.A.Fenn@sussex.ac.uk

Colin Rourke
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: cpr@maths.warwick.ac.uk

Brian Sanderson
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: bjs@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03912-2
PII: S 0002-9947(06)03912-2
Keywords: Classifying space, codimension 2, cubical set, James bundle, link, knot, $\pi_2$, rack
Received by editor(s): August 1, 2003
Received by editor(s) in revised form: November 24, 2004
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.