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Central Adams operators. (English) Zbl 0638.55005

“From the introduction: “Let X be a (pointed) finite complex with trivial rational homology. Then, for some, potentially large, integer \(s>0\) there exists a stable map \(B: \Sigma\) \({}^ sX\to X\) which induces an isomorphism \(B^*: KO^*(X)\to KO^*(\Sigma^ sX)\) in (reduced) periodic real K-theory. Such maps were first considered by Adams and Toda. Recent work of Devinatz, Hopkins and Smith shows, as a special first case of a much more general theorem, that B can be chosen to be central in the graded ring \(\pi^*_ s\{X;X\}\) of stable self-maps of X. In this note we reprove this using the elementary methods, based on a solution of the Adams conjecture, that we described in Topology 24, 475- 486 (1985; Zbl 0581.55008). The key idea comes from M. Hopkins [Lectures at the Durham Sympos. on Homotopy Theory (1985)].”
Reviewer: M.Mimura

MSC:

55N15 Topological \(K\)-theory
55P42 Stable homotopy theory, spectra
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
55Q10 Stable homotopy groups

Citations:

Zbl 0581.55008
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References:

[1] ADAMS, J.F.: On the groups J(X) - IV. Topology5, 21-71 (1966) · Zbl 0145.19902 · doi:10.1016/0040-9383(66)90004-8
[2] CRABB, M.C. and KNAPP, K.: Adams periodicity in stable homotopy. Topology24, 475-486 (1985) · Zbl 0581.55008 · doi:10.1016/0040-9383(85)90016-3
[3] CRABB, M.C. and KNAPP, K.: The Hurewicz map on stunted complex projective spaces. Preprint (1986) · Zbl 0665.55007
[4] CRABB, M.C. and KNAPP, K.: Adams trivialization, Im(J)-theory and the codegree of vector bundles. In preparation
[5] CRABB, M.C. and SUTHERLAND, W.A.: The space of sections of a sphere-bundle I. Proc. Edinburgh Math. Soc.29, 383-403 (1986) · Zbl 0614.55012 · doi:10.1017/S0013091500017831
[6] HOPKINS, M.: Lectures at the Durham Symposium on Homotopy Theory 1985
[7] SEYMOUR, R.M.: Vector bundles invariant under the Adams operations. Quart. J. Math. Oxford (2)25, 395-414 (1974) · Zbl 0302.55009 · doi:10.1093/qmath/25.1.395
[8] TODA, H.: Order of the identity class of a suspension space. Annals of Math.78, 300-325 (1963) · Zbl 0146.18901 · doi:10.2307/1970345
[9] WATANABE, T.: On the spectrum representing algebraic K-theory for a finite field. Osaka J. Math.22, 447-462 (1985) · Zbl 0585.55001
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