Celebratio Mathematica

R. Fenn, C. Rourke, and B. Sanderson: “Trunks and classifying spaces,” Appl. Categ. Struct. 3 : 4 (1995), pp. 321–356. MR 1364012 Zbl 0853.55021 article

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks. A rack \( X \) gives rise to a trunk \( \mathcal{T} ( X ) \) which has a single vertex and the set \( X \) as set of edges. The rack space \( B X \) of \( X \) is the realisation of the nerve \( N \mathcal{T} ( X ) \) of \( \mathcal{T} ( X ) \). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural map \( B X \to B As ( X ) \) where \( B As ( X ) \) is the classifying space of the associated group of X. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown–Higgins classifying space of the associated crossed module [Fenn and Rourke 1992] and [Brown and Higgins 1991]. The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.

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R. Fenn, C. Rourke, and B. Sanderson: “James bundles,” Proc. Lond. Math. Soc. (3) 89 : 1 (2004), pp. 217–240. MR 2063665 Zbl 1055.55005 article

We study cubical sets without degeneracies, which we call \( \square \)-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a \( \square \)-set \( C \) has an infinite family of associated \( \square \)-sets \( J^i (C), \) for \( i = 1, 2,\dots \), which we call James complexes. There are mock bundle projections \[ p_i: |J^i(C)|\to |C| \] (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of \( \Omega (S^2) \). The algebra of these classes mimics the algebra of the cohomotopy of \( \Omega (S^2) \) and the reduction to cohomology defines a sequence of natural characteristic classes for a \( \square \)-set. An associated map to \( BO \) leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.

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R. Fenn, C. Rourke, and B. Sanderson: “The rack space,” Trans. Amer. Math. Soc. 359 : 2 (2007), pp. 701–740. MR 2255194 Zbl 1123.55006 article

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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Colin P. Rourke

Colin Rourke

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