𝑆¹-equivariant function spaces and characteristic classes

Mann, Benjamin M.; Miller, Edward Y.; Miller, Haynes R.

doi:10.1090/S0002-9947-1986-0831198-6

$S^ 1$-equivariant function spaces and characteristic classes
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by Benjamin M. Mann, Edward Y. Miller and Haynes R. Miller PDF

Trans. Amer. Math. Soc. 295 (1986), 233-256 Request permission

Abstract:

We determine the structure of the homology of the Becker-Schultz space $SG({S^1}) \simeq Q({\mathbf {C}}P_ + ^\infty \wedge {S^1})$ of stable ${S^1}$-equivariant self-maps of spheres (with standard free ${S^1}$-action) as a Hopf algebra over the Dyer-Lashof algebra. We use this to compute the homology of $BSG({S^1})$. 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 "${S^1}$-transfers" ${\mathbf {C}}P_ + ^\infty \wedge {S^1} \to Q((B{{\mathbf {Z}}_{{p^n}}})_+ ),n \geq 0$, and of the equivariant $J$-homomorphisms $SO \to Q({\mathbf {R}}P_ + ^\infty )$ and $U \to Q({\mathbf {C}}P_ + ^\infty \wedge {S^1})$. By composing, we obtain $U \to Q{S^0}$ in homology, answering a question of J. P. May.

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