James Bundles and Applications

SUMMARY We study cubical sets without degeneracies, which we call {sets. These sets arise naturally in a number of settings and there is a beautiful geometry associated with them; in particular a {set C has an innnite family of associated {sets J i (C), i = 1; 2; : : :, which we call James complexes. There are projections p i : jJ i (C)j ! jCj which are mock bundle projections. These deene classes in unstable cohomotopy which generalise the classical James-Hopf invariants of (S 2) and for this reason we call these bundles James bundles. The algebra of James bundles mimics the algebra of the cohomotopy of (S 2) and the reduction to cohomology deenes a sequence of natural characteristic classes for a {set. The main application is to the rack space (deened in FRS]). In this case the rst James bundle is a classifying bundle for links of codimension 2 with representation in the given rack. The invariants of the rack space (including the James{Hopf invariants) provide many new invariants for racks and hence for knots and links in codimension 2. A second application, which will be explored further in a later paper is to Vassiliev theory. There is a deenition of a Vassiliev invariant for any {set generalising the classical case and an obstruction theory with obstructions in the cohomology of the James complexes for extending a partially deened invariant over the whole set. Other applications are given. There is a combinatorial interpretation for 2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3{sphere. There is also a generalised cohomology theory associated to any {set with geometric interpretation similar to that for Mahowald orientation M, S1].