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The rack space

Roger Fenn·Colin Rourke·Brian Sanderson
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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 (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bundles" math.AT/0301354). 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 π_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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Roger Fenn, Colin Rourke, Brian Sanderson. 2003-06-15. The rack space. https://arxiv.org/abs/math/0304228

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