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; 40(1):43-94. 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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    • "We are mostly motivated by the applications of our constructions to dg-algebras over the operads P = Com, E , because these categories of dg-algebras define models for the homotopy of spaces (rationally or completed at a prime depending on the context). We mostly refer to [11] [12] for these applications of dg-algebras in homotopy theory. Let A be an object in any of our categories of dg-algebras. "
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    ABSTRACT: We consider the cotriple resolution of algebras over operads in differential graded modules. We focus, to be more precise, on the example of algebras over the differential graded Barratt-Eccles operad and on the example of commutative alegbras. We prove that the geometric realization of the cotriple resolution (in the sense of model categories) gives a cofibrant resolution functor on these categories of differential graded algebras.
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    • "There is also a significant amount of unpublished work due, independently, to Jim Milgram and Ezra Getzler to be acknowledged. These constructions are significant since by [8] [9], under suitable assumptions on Y , an E ∞ -algebra structure on C * (Y ) determines the homotopy type of Y . Perhaps the main theme in this paper is to understand better the relation between chain cooperations (or cochain operations) and the combinatorics of joins. "
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    ABSTRACT: The join operad arises from the combinatorial study of the iterated join of simplices. We study a suitable simplicial version of this operad which includes the symmetries given by permutations of the factors of the join. From this combinatorics we construct an E-infinity operad which coacts naturally on the chains of a simplicial set.
    HOMOLOGY HOMOTOPY AND APPLICATIONS 10/2011; 15(2). DOI:10.4310/HHA.2013.v15.n2.a15 · 0.36 Impact Factor
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    • "For example, the Auslander-Reiten quiver of a triangulated category is an interesting combinatorial invariant; see [15], [16], [18], [19] and [35]. The singular (co)chain complex functor is a necessary ingredient in developing algebraic model theory for topological spaces; see [1], [3], [10], [14] and [29]. We will here advertise the idea that the functor, combined with tools from categorical representation theory of kind just mentioned, is likely to provide new insights into the relationship between algebra and topology. "
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    ABSTRACT: The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover we provide a method to compute the invariant for spaces over a $\K$-formal space. This enables us to determine the level of the total space of a bundle over the 4-dimensional sphere with the aid of Auslander-Reiten theory for spaces due to J{\o}rgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization.
    Journal of Pure and Applied Algebra 02/2011; 216(4). DOI:10.1016/j.jpaa.2011.08.009 · 0.58 Impact Factor
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