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: We consider the Quillen adjunction between fixed points and inflation in the context of equivariant module spectra over equivariant ring spectra, and give numerous examples including some based on geometric fixed points and some on the Eilenberg-Moore spectral sequence. These results were originally presented as part of our equivalence between rational torus-equivariant spectra and an algebraic model in arXiv:1101.2511. However, the present results apply in many other interesting cases explored here, which are not rational and where the ambient group is not a torus. The material in arXiv:1101.2511v3 will be revised to refer to this paper.
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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 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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