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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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