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Register a new userWhitehead products in function spaces: Quillen model formulae
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We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map f: X to Y, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of f. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components.
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There exists a canonical functor from the category of fibrant objects of a model category modulo cylinder homotopy to its homotopy category. We show that this functor is faithful under certain conditions, but not in general.
There are two main approaches to the problem of realizing a $Pi$-algebra (a graded group $Lambda$ equipped with an action of the primary homotopy operations) as the homotopy groups of a space $X$. Both involve trying to realize an algebraic free simplicial resolution $G_bullet$ of $Lambda$ by a simplicial space $W_bullet$ and proceed by induction on the simplicial dimension. The first provides a sequence of Andr{e}-Quillen cohomology classes in $H_{AQ}^{n+2}(Lambda;Omega^{n}Lambda)$ for $n geq 1$ as obstructions to the existence of successive Postnikov sections for $W_bullet$ by work of Dwyer, Kan and Stover. The second gives a sequence of geometrically defined higher homotopy operations as the obstructions by earlier work of Blanc; these were identified with the obstruction theory of Dwyer, Kan and Smith in earlier work of the current authors. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of $Lambda$. In this paper we 1) provide an explicit construction of the cocycles representing the cohomology obstructions; 2) provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to long Toda brackets), and 3) show that these two constructions correspond under an evident map.
We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v_n-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.
We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.
The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying space of a finite group, and whose fibre is the classifying space of a p-local compact group, the total space is, up to p-completion the classifying space of a p-local compact group.
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