E∞ algebras and p-adic homotopy theory

Department of Mathematics, Massachusetts Institute of Technology, Room 2-265, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Topology (Impact Factor: 0.23). 01/2001; DOI: 10.1016/S0040-9383(99)00053-1

ABSTRACT Let denote the field with p elements and its algebraic closure. We show that the singular cochain functor with coefficients in induces a contravariant equivalence between the homotopy category of connected p-complete nilpotent spaces of finite p-type and a full subcategory of the homotopy category of -algebras.

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    ABSTRACT: In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as G-spaces and E_n-ring spectra. In this paper we study special cases of this spectral sequence in detail. In special cases, we show that the Goerss-Hopkins spectral sequence and the T-algebra spectral sequence agree. Applying a result of Jennifer French, which we extend to the rational case, these spectral sequences agree with the unstable Adams spectral sequence after further restriction. From these equivalences we obtain information about filtration and differentials. Using these equivalences we construct the homological and cohomological Bockstein spectral sequences topologically. We apply theses spectral sequences to show that Hirzebruch genera can be lifted to E_\infty-ring maps and that the forgetful functor from E_\infty-algebras in H\fpbar-modules to H_\infty-algebras is neither full nor faithful.
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    ABSTRACT: We give a source of examples of H_infinity ring structures that do not lift to E_infinity ring structures, based on Mandell's equivalence between certain cochain algebras and spaces.
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    ABSTRACT: In the paper under review the author proves a Koszul duality result for E n -operads in differential graded modules over a ring. In the case n=1, an E 1 -operad is equivalent to the associative operad, and the result is classical. For n>1, the homology of an E n -operad E n is identified with the operad of n-Gerstenhaber algebras: H * (E n )=G n . The operad G n is Koszul, and its dual K(G) is the operadic n-fold desuspension of the cooperad G n ∨ dual to G n in ℤ-modules: K(G n )=Λ -n G n ∨ , where Λ is the operadic suspension operation, and (-) ∨ is the duality of ℤ-modules. Thus there is a weak-equivalence ϵ n :B c (Λ -n G n ∨ )→G n in the category of differential graded operads, where B c (D) is the cobar construction on a cooperad D. For the characteristic zero setting, this yields a Koszul duality result. But, for ℤ-modules it is necessary to introduce new ideas. The author shows that the weak-equivalence ϵ n is realized by a morphism at the chain level ψ n :B c (Λ -n E n ∨ )→E n . To state the realization condition properly, the author considers a natural spectral sequence E 1 =B c (H * (Λ -n E n ∨ ))⇒H * (B c (Λ -n E n ∨ )) associated with the cobar construction. Then the author shows that the restriction of the homology morphism induced by ψ n on the edge of the spectral sequence agrees with ϵ n for n>1.
    Selecta Mathematica 01/2011; 17(2). · 0.84 Impact Factor


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