Homotopy orbits of free loop spaces

ArticleOriginal scientific text

Title

Authors ¹, ²

Department of Mathematics, University of Aarhus, Ny Munkegade, DK-8000 Århus C, Denmark
Département de Mathématiques, Université de Paris Nord, URA 742 du CNRS, Avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France

Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors

Ω_{λ} (R)

and ℓ(R) together with algebra homomorphisms

e : Ω_{λ} (R) \to H \cdot (Λ X)

and

ψ : ℓ (R) \to H \cdot (E S^{1} \times_{S^{1}} Λ X)

. When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

[Loday] J.-L. Loday, Cyclic Homology, Grundlehren Math. Wiss. 301, Springer, 1992.
[Madsen] I. Madsen, Algebraic K-theory and traces, in: Current Developments in Mathematics, International Press, Boston, 1995, 191-321.
[Milgram] R. J. Milgram, Unstable Homotopy from the Stable Point of View, Lecture Notes in Math. 368, Springer, 1974.
[Ottosen] I. Ottosen, Higher cyclic reduced powers, J. Pure Appl. Algebra, to appear.
[Schwartz] L. Schwartz, Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture, Chicago Lectures in Mathematics, Univ. of Chicago Press, 1994.
[Smith81] L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103 (1981), 887-910.
[Smith84] L. Smith, The Eilenberg-Moore spectral sequence and the mod 2 cohomology of certain free loop spaces, Illinois J. Math. 28 (1984), 516-522.
[St-Ep] N. E. Steenrod and D. B. A. Epstein, Cohomology Operations, Ann. of Math. Stud. 50, Princeton Univ. Press, 1962.