Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ S\sp 1$-equivariant function spaces and characteristic classes

Authors: Benjamin M. Mann, Edward Y. Miller and Haynes R. Miller
Journal: Trans. Amer. Math. Soc. 295 (1986), 233-256
MSC: Primary 55R40; Secondary 55Q50, 55R12, 55R91
MathSciNet review: 831198
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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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Article copyright: © Copyright 1986 American Mathematical Society