## 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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