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# S1-Equivariant Function Spaces and Characteristic Classes

Benjamin M. Mann, Edward Y. Miller and Haynes R. Miller
Transactions of the American Mathematical Society
Vol. 295, No. 1 (May, 1986), pp. 233-256
DOI: 10.2307/2000155
Stable URL: http://www.jstor.org/stable/2000155
Page Count: 24
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## Abstract

We determine the structure of the homology of the Becker-Schultz space $SG(S^1) \simeq Q(CP_+^\infty \wedge S^1)$ of stable S1-equivariant self-maps of spheres (with standard free S1-action) as a Hopf algebra over the Dyer-Lashof algebra. We use this to compute the homology of BSG(S1). Along the way, we give a fresh account of the partially framed transfer construction and the Becker-Schultz homotopy equivalence. We compute the effect in homology of the "S1-transfers" $CP_+^\infty \wedge S^1 \rightarrow Q((BZ_pn)_+), n \geq 0$, and of the equivariant J-homomorphisms SO → Q(RP+ ∞) and $U \rightarrow Q(P_+^\infty \wedge S^1)$. By composing, we obtain U → QS0 in homology, answering a question of J. P. May.

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