k\(<n>\)-theories of bordisms with singularities and \(k<n>\)-orientability of fibre bundles. (Russian) Zbl 0547.55001
There are two main results in this paper. The first gives a description of the homotopy types of the spectra \(k<n>\) corresponding to bordism theories with singularities for which \(\pi_*(k<n>)=({\mathbb{Z}}/p)[t]\) with dim t\(=2(p^ n-1)\) for a prime p. The main tool used is the Postnikov tower of such a spectrum \(k<n>\), together with the corresponding invariants which are higher-order cohomology operations denoted \(\tilde Q_ n^{(s)}\). The second result gives a necessary and sufficient condition for a vector bundle or sphere bundle to be \(k<n>\)-orientable, namely all the operations \(\tilde Q_ n^{(s)}\) must act trivially on the Thom class of the bundle.
Reviewer: P.Landweber
MSC:
55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |
55R25 | Sphere bundles and vector bundles in algebraic topology |
55S20 | Secondary and higher cohomology operations in algebraic topology |