By Gunnar Carlsson, Ralph Cohen, Haynes R. Miller, Douglas C. Ravenel

Those are lawsuits of a world convention on Algebraic Topology, held 28 July via 1 August, 1986, at Arcata, California. The convention served partially to mark the twenty fifth anniversary of the magazine *Topology* and sixtieth birthday of Edgar H. Brown. It preceded ICM 86 in Berkeley, and used to be conceived as a successor to the Aarhus meetings of 1978 and 1982. a few thirty papers are integrated during this quantity, typically at a examine point. topics comprise cyclic homology, H-spaces, transformation teams, genuine and rational homotopy thought, acyclic manifolds, the homotopy thought of classifying areas, instantons and loop areas, and intricate bordism.

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**Example text**

X] homotopy theory, Springer Lecture Notes 100 (1969). 51 52 We apply the formal completion to rational homotopy types where profinite completion gives only contractible spaces. In this case, an essential ingredient in the extra topological struc¯ is a Z-module structure on the homotopy ture on the functor [ , X] ¯ This Z-structure allows one to treat these groups which groups of X. are enormous Q-vector spaces. These completion constructions and the localization of Chapter 2 are employed to fracture a classical homotopy type into one rational and infinitely many p-adic pieces.

Z← −Z← − Z) = ← p=2 If we consider mixing these operations in groups, the following remarks are appropriate. i) localizing and then profinitely completing is simple and often gives zero, Gp if ∩ = ∅ (G ) = p∈ ∩ 0 if ∩ = ∅ . g. (G, l, ) = (Z , ∅, p) gives Qp = 0. g. ¯ p = Q ⊗ Zp , the “field of p-adic numbers”, usually a) (Z0 )¯p = Q denoted by Qp . Qp is the field of quotients of Zp (although it 27 Algebraic Constructions is not much larger because only 1/p has to be added to Zp to make it a field).

Using the natural sequence (over Z ) 0 → Ext Hi ( ; Z ), Z → H i+1 ( ;Z ) → Hom Hi+1 ( ; Z ), Z →0 we see that induces an isomorphism of Z -cohomology. By universal coefficients (over Z ) the obstruction groups all vanish. Thus there is a unique extension f , and is a localization. Chapter 3 COMPLETIONS IN HOMOTOPY THEORY In this Chapter we extend the completion constructions for groups to homotopy theory. In spirit we follow Artin and Mazur1 , who first conceived of the profinite completion of a homotopy type as an inverse system of homotopy types with finite homotopy groups.