Abstract
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 Ji(C), for i = 1, 2, …, which we call James complexes. There are mock bundle projections pi: |Ji(C)| → |C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of Ω(S2). The algebra of these classes mimics the algebra of the cohomotopy of Ω(S2) 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).
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