James bundles

We study cubical sets without degeneracies, which we call □‐sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a □‐set C has an infinite family of associated □‐sets J i ( C ), for i = 1, 2, …, which we call James complexes. There are mock bundle projections p i : | J i ( C )| → | C | (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of Ω( S 2 ). The algebra of these classes mimics the algebra of the cohomotopy of Ω( S 2 ) and the reduction to cohomology defines a sequence of natural characteristic classes for a □‐set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation. 2000 Mathematics Subject Classification 55N22, 55P44 (primary), 57R15, 57R20, 57R90 (secondary).