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On PL DeRham Theory and Rational Homotopy Type
The rational [bold]PL de Rham theory of Sullivan is developed and generalized, using methods of Quillen's "homotopical algebra." For a field k of characteristic 0, a pair of contravariant adjoint functors A : (Simplicial sets) [right arrow over left arrow] (Commutative DG k-algebras) : F is obtained which pass to the appropriate homotopy categories. When k is the field of rationals, these functors induce equivalence between the appropriate simplicial and algebraic rational homotopy categories. The theory is not restricted to simply connected spaces. It is closely related to the theory of "rational localization" (for nilpotent spaces) and "rational completion" in general.
Homotopy Limits, Completions and Localizations
The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan; for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others. In part II of these notes, the authors have assembled some results on towers of fibrations, cosimplicial spaces and homotopy limits which were needed in the discussions of part I, but which are of some interest in themselves.
Homological Localization Towers for Groups and Pi-Modules
This paper investigates algebraic homological localizations which have arisen in the homotopy theoretic study of localizations of spaces. It focuses on the "HR-localization" of groups and the "HZ localization" of modules over a group [capital Greek]Pi, where the coefficient ring [italic]R is allowed to be a subring of the rationals or a finite cyclic ring. The investigation is based on a construction of natural transfinite towers which eventually stabilize to the desired homological localizations. These towers are used to show that the HR-local groups form the smallest class of groups containing the trivial group, closed under inverse limits, and closed under central [italic]R-module extensions.
National Union Catalog
Includes entries for maps and atlases.
