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\(S^ 1\)-equivariant function spaces and characteristic classes. (English) Zbl 0597.55010
Let H be a compact Lie group with a finite dimensional orthogonal representation W such that H acts freely on the unit sphere sW. Let \(G(H)=\lim_{\vec n}End_ H(s(nW))\) and let SG(H) be the component of G(H) which contains the identity map. J. C. Becker and R. E. Schultz [Comment. Math. Helv. 49, 1-34 (1974; Zbl 0278.55006)] proved that \(G(H)\simeq Q(BH^{\zeta})\) for an appropriate vector bundle \(\zeta\). In particular \(SG(S^ 1)\simeq Q(CP_+^{\infty}\wedge S^ 1)\). Let p be a prime. The authors compute \(H_*(SG(S^ 1);Z_ p)\) as a Hopf algebra over the Dyer-Lashof algebra. Then they use the Eilenberg- Moore spectral sequence to compute \(H_*(BSG(S^ 1);Z_ p)\). They also compute the effect in mod p homology of the \(S^ 1\)-transfers \(CP_+^{\infty}\wedge S^ 1\to Q(BZ_{p^ n}+)\), the equivariant J- homomorphisms \(SO\to Q(RP_+^{\infty})\), \(U\to Q(CP_+^{\infty}\wedge S^ 1)\) and the classical J-homomorphism \(U\to QS^ 0\). The latter computation solves a problem of J. P. May [F. R. Cohen, T. J. Lada and J. P. May, The homology of iterated loop spaces (Lect. Notes Math. 533) (1976; Zbl 0334.55009), p. 123].
Reviewer: S.Kochman

MSC:
55P91 Equivariant homotopy theory in algebraic topology
55P47 Infinite loop spaces
55Q50 \(J\)-morphism
55Q91 Equivariant homotopy groups
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