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Quelques éléments dans l’homotopie stable du groupe unitaire. (Some elements in the stable homotopy of the unitary group). (French) Zbl 0638.55010

Let U(m) be the unitary group and \(\lambda_ i\) a generator of the primitives in \(H_{2i+1}(U(m))\), \(i\leq m\). Essentially as a consequence of Bott periodicity it is known that \(\delta_ i:=(i!)\lambda_ i\) is spherical, giving a complete description of the unstable Hurewicz map \(h: \pi\) \({}_*(U(m))\to H_*(U(m))\). In contrast to the unstable case not much is known on the image of the stable Hurewicz map \(h^ s: \pi_*^ s(U(m))\to H_ m(U(m))\) beyond the information coming from im(h). The aim of this paper is to obtain information on \(im(h^ s).\)
For a finite CW-complex C embedded in \(S^{n+1}\) consider a regular neighbourhood with boundary M. Then M is a hypersurface and together with a map \(f: M\to U(m)\), m large, this determines an element in \(\pi_ n^ s(U(m))\) (via the Thom-Pontryagin construction; actually in \(\pi_{n+1}(\Sigma U(m))\) since M is a hypersurface). Using this construction with \(C=cofibre\) of \(\alpha_{k/t}\) and \(\alpha_{k/t}\) an element in \(\pi_{2\cdot (k(p-1)+t)}(S^{2t+1})\) with e-invariant \(1/p^ t\), p an odd prime, the author defines families of hypersurfaces realizing nontrivial elements in the image of \(h^ s: \pi_*^ s(U(m))\to H_*(U(m))\). For example, the elements \(\delta_{i,j}=(\delta_ i\cdot \delta_{j+p-1}-\delta_{i+p-1}\cdot \delta_ j)/p\) and \(\delta_ i\cdot \delta_{i+p^{t-1}(p-1)}/p^ t\) in \(im(h^ s)\) are obtained in this way. Using a p-local splitting of U(m) a complete computation of \(im(h^ s)\) for \(m\leq 2p-1\), \(p\neq 2\), is obtained: the elements \(\delta_ i\) and \(\delta_{i,j}\) and their various products generate \(im(h^ s).\)
In the last chapter the author investigates the next interesting case U(m), \(m\geq 2p\). He constructs a hypersurface realizing \(\delta_ 1\cdot \delta_ p\cdot \delta_{2p-1}/p^ 2\) in \(im(h^ s)\). This paper is a continuation of the author’s work with L. Schwartz [C. R. Acad. Sci., Paris, Sér. I 299, 619-622 (1984; Zbl 0571.55011)]. It also contains detailed proofs for some of the results of that note.
Reviewer: K.H.Knapp

MSC:

55Q10 Stable homotopy groups
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57T10 Homology and cohomology of Lie groups
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory

Citations:

Zbl 0571.55011
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References:

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