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James bundles. (English) Zbl 1055.55005

The present paper performs a wide and accurate study of the so called square sets, which are cubical sets without degeneracies, first introduced by the same authors in [Appl. Categ. Struct. 3, No. 4, 321–356 (1995; Zbl 0853.55021)] through the concept of a trunk.
In particular, for each square set \(C\), the authors introduce an infinite family of associated square sets \(J^i(C)\), for \(i\in \mathbb N\), which are called James complexes.
On the other hand, James complexes \(J^i(C)\) define a sequence of mock bundles with base \(C\), called James bundles, which have a have a strong connection with classical James-Hopf invariants: in fact, they define classes in unstable cohomotopy which generalize classical James-Hopf invariants of \(\Omega(S^2)\).
Further, the paper examines the algebra generated by the James classes in stable cohomotopy and in cohomology; natural characteristic classes for a square set are so defined, whose mod 2 reductions are pulled back from both the Stiefel-Whitney and the Wu classes of BO.
Finally, the map to BO is used to define the concept of a square-set orientation for a manifold; this leads to a generalized (co)-homology theory, which has a geometric interpretation close to that for Mahowald orientation [see M. Mahowald, Duke Math. J. 46, 549–559 (1979; Zbl 0418.55012)] given by the third author in [Algebraic topology, Proc. Symp., Aarhus 1978, Lect. Notes Math. 763, 152–174 (1979; Zbl 0423.57011)].

MSC:

55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R20 Characteristic classes and numbers in differential topology
57R90 Other types of cobordism
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