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Fenn, Roger, Rourke, C. P. and Sanderson, B. J.. (2004) James bundles. Proceedings of the London Mathematical Society, Vol.89 (No.1). pp. 217240. ISSN 00246115

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Official URL: http://dx.doi.org/10.1112/S0024611504014674
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 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 {Omega}(S2). The algebra of these classes mimics the algebra of the cohomotopy of {Omega}(S2) 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.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Algebraic topology, Manifolds (Mathematics), Differentiable manifolds, Cobordism theory, Differential topology 
Journal or Publication Title:  Proceedings of the London Mathematical Society 
Publisher:  Cambridge University Press 
ISSN:  00246115 
Official Date:  July 2004 
Volume:  Vol.89 
Number:  No.1 
Page Range:  pp. 217240 
Identification Number:  10.1112/S0024611504014674 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. R. ANTOLINI, ‘Cubical structures and homotopy theory’, PhD Thesis, University of Warwick, 1996. 
URI:  http://wrap.warwick.ac.uk/id/eprint/751 
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