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James bundles

Published online by Cambridge University Press:  30 June 2004

Roger Fenn
Affiliation:
Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH. E-mail: R.A.Fenn@sussex.ac.uk
Colin Rourke
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL United Kingdom. E-mail: cpr@maths.warwick.ac.uk
Brian Sanderson
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL United Kingdom. E-mail: bjs@maths.warwick.ac.uk
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Abstract

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 $i=1,2,\ldots$, which we call James complexes. There are mock bundle projections $p_i \colon |J^i (C)| \to |C|$ (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.

Type
Research Article
Copyright
2004 London Mathematical Society

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