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James bundles

By Roger Fenn, C. P. Rourke and B. J. Sanderson


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

Topics: QA
Publisher: Cambridge University Press
Year: 2004
OAI identifier:

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  1. (2004). A classi cation of classical links’,
  2. (1982). A classifying invariant of knots; the knot quandle’, doi
  3. (1976). A geometric approach to homologytheory ,
  4. (1958). A surveyof binarysy stems (Springer,
  5. (1997). and cubical sets’, PhD Thesis,
  6. (1990). Cohomology of knotspaces’, Theoryof singularities and its applications
  7. (1981). Commutator calculus and groups of homotopyclasses , doi
  8. (1996). Cubical structures and homotopy theory’, doi
  9. (1992). Distributivgesetze in der Homotopietheorie’, PhD Thesis,
  10. (1978). KðZ=2Þ as a Thom spectrum’,
  11. (1956). On the suspension triad’, doi
  12. (1995). On the Vassiliev knot invariants’, doi
  13. (1992). Racks and links in codimension two’, doi
  14. (1955). Reduced productspaces’,
  15. (1979). Ring spectra which are Thom complexes’, doi
  16. (1978). Self-intersections and higher Hopf invariants’, doi
  17. (1959). Self-intersections of immersed manifolds’, doi
  18. (1971). sets, I: homotopy theory’, doi
  19. (1969). Some well known weak homotopy equivalences are genuine homotopy equivalences’, INDAM,
  20. (1978). The geometry of Mahowald orientations’, Algebraic topology, Aarhus, doi
  21. (2003). The rack space’,
  22. (1999). The singular cubical setof a t opological space’,
  23. (1995). Trunks and classifying spaces’,

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