## The rack space

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- by Roger Fenn, Colin Rourke and Brian Sanderson PDF
- Trans. Amer. Math. Soc.
**359**(2007), 701-740 Request permission

## 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.## References

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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
- Received by editor(s): August 1, 2003
- Received by editor(s) in revised form: November 24, 2004
- Published electronically: August 24, 2006
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 701-740 - MSC (2000): Primary 55Q40, 57M25; Secondary 57Q45, 57R15, 57R20, 57R40
- DOI: https://doi.org/10.1090/S0002-9947-06-03912-2
- MathSciNet review: 2255194