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Construction d’éléments dans \(\pi^ s_*(BU(2))\). (French) Zbl 0547.55011

The authors give a geometric construction of elements belonging to \(\pi^ s_*(BU(2))\) which are not in the image of \(\pi^ s_*(B{\mathbb{T}}^ 2)\) via the map induced by the inclusion \({\mathbb{T}}^ 2\hookrightarrow u(2)\) of a maximal torus. This contrasts with the situation for a complex oriented theory, where the map is subjective.
The construction is of a family \(X_{4k+1}\) of dimension 16 k\(+6\), \(k\geq 0\), and proceeds by equivariant and surgery methods. The failure of \(X_{4k+1}\) to arise from \(B{\mathbb{T}}^ 2\) is shown by computing its KU theoretic Hurewicz image.

MSC:

55Q10 Stable homotopy groups
55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R90 Other types of cobordism
57T20 Homotopy groups of topological groups and homogeneous spaces
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References:

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