James bundles

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©, i=1,2,…, which we call James complexes. There are mock bundle projections p_i:|J^i©|–>|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 [M Mahowald, Ring Spectra which are Thom complexes, Duke Math. J. 46 (1979) 549–559] and [B Sanderson, The geometry of Mahowald orientations, SLN 763 (1978) 152–174].