By J. P. May

This quantity introduces equivariant homotopy, homology, and cohomology concept, in addition to a variety of comparable issues in smooth algebraic topology. It explains the most rules in the back of one of the most awesome fresh advances within the topic. The e-book starts with a improvement of the equivariant algebraic topology of areas culminating in a dialogue of the Sullivan conjecture that emphasizes its dating with classical Smith idea. It then introduces equivariant reliable homotopy conception, the equivariant strong homotopy classification, and an important examples of equivariant cohomology theories. the elemental equipment that's had to make critical use of equivariant strong homotopy thought is gifted subsequent, in addition to discussions of the Segal conjecture and generalized Tate cohomology. ultimately, the ebook offers an creation to ``brave new algebra'', the examine of point-set point algebraic buildings on spectra and its equivariant functions. Emphasis is put on equivariant complicated cobordism, and similar effects on that subject are awarded intimately. beneficial properties: Introduces some of the basic rules and ideas of recent algebraic topology. offers entire fabric now not present in the other e-book at the topic. offers a coherent evaluation of many parts of present curiosity in algebraic topology. Surveys loads of fabric, explaining major rules with out getting slowed down in information.

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**Example text**

C) Each H : X H ?! (X^p)H is completion at p. For n 1, there is a natural short exact sequence 0 ?! L0 n (X ) ?! n(X^p) ?! 1(X ) ?! 0: Proofs. 2, except that completions of Eilenberg-Mac Lane G-spaces are not Eilenberg-Mac Lane G-spaces in general. For a coe cient system M , : M ?! L0M induces p-completions K (M; n) ?! K (L0M; n) whenever L1M = 0. For the general case, let FM be 32 II. POSTNIKOV SYSTEMS, LOCALIZATION, AND COMPLETION the coe cient system obtained by applying the free Abelian group functor to M regarded as a set-valued functor.

The advantage of doing so is that rational homotopy theory is determined completely by algebraic invariants, as was shown by Quillen and later by Sullivan. Our theory is analogous to Sullivan's theory of minimal models, which we now review. For our purposes we prefer Sullivan's approach because of its computational advantage and its relation to geometry by use of di erential forms. The algebraic invariants that determine the rational homotopy type are certain algebras that we call DGA's. By de nition a DGA is a graded, commutative, associative algebra with unit over the rationals, with di erential d : An !

The quotient map G ?! G=H induces a map : T (G=H ) ?! T (G=e)H , and these maps together specify a natural map : T ?! T . Passage from : T ?! X to : T ?! X is a bijection whose inverse sends f : T ?! X to f . The following result of Elmendorf shows that G -spaces determine G-spaces in a less obvious way. In fact, up to homotopy, any G -space can be realized as the xed point system of a G-space and, up to homotopy, the functor has a right adjoint as well as a left adjoint. Note that we can form the product T K of a G -space 3.