James bundles
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), 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 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].
Cite
@article{arxiv.math/0301354,
title = {James bundles},
author = {Roger Fenn and Colin Rourke and Brian Sanderson},
journal= {arXiv preprint arXiv:math/0301354},
year = {2007}
}
Comments
This paper is extracted from our January 1996 preprint `James bundles and applications' available at: http://www.maths.warwick.ac.uk/~cpr/ftp/james.ps