Celebratio Mathematica — Rourke

My collaboration with Colin Rourke

Works connected to Roger Fenn

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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.

Roger Fenn	Related
Brian Joseph Sanderson	Related

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.

Roger Fenn	Related
Brian Joseph Sanderson	Related