By Barcelona Conference on Algebraic Topology 1990 San Feliu De Guixols, Manuel Castellet, J. Aguade, Frederick R. Cohen

The papers during this assortment, all absolutely refereed, unique papers, replicate many features of contemporary major advances in homotopy conception and workforce cohomology. From the Contents: A. Adem: at the geometry and cohomology of finite easy groups.- D.J. Benson: Resolutions and Poincar duality for finite groups.- C. Broto and S. Zarati: On sub-A*-algebras of H*V.- M.J. Hopkins, N.J. Kuhn, D.C. Ravenel: Morava K-theories of classifying areas and generalized characters for finite groups.- okay. Ishiguro: Classifying areas of compact easy lie teams and p-tori.- A.T. Lundell: Concise tables of James numbers and a few homotopyof classical Lie teams and linked homogeneous spaces.- J.R. Martino: Anexample of a solid splitting: the classifying area of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy specialty of BU(2) at the top 2.- G. Mislin: Cohomologically primary components and fusion in teams.

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**Sample text**

We call such groups generically trivial. In Sections 1 and 2 we exhibit their basic properties and point out several sources of examples. A group G is called separable if for some family of idempotent functors {Ep) a~ above, the canonical homomorphism from G to the cartesian product of the groups EpG is injective. Residually nilpotent groups and many others are separable. (X, Y) of pointed maps from X to Y is weakly contractible for a very broad class of spaces Y, namely for all those Y such that ~rl(Y ) is separable.

9 P r o p o s i t i o n . [Compare with [S]] An unstable A~-module N is m-nilpoten¢ if and only if f~kN is nilpotent for any k: 0 < k < m. - Suppose N m-nilpotent and k: 0 < k < m. The natural surjection N --~ Ekf/kN shows that E k ~ k N is m-nilpotent. 8 proves that f~kN is (m - k)-nilpotent, so, in particular, it is nilpotent. To prove the converse we use induction on m and the Mahowald exact sequence (E2) for the Brown-Gitler modules J ( n ) , n > O. 10 Remark: Let Sqk, k E Z, the operation defind by Sqkx = Sql*l-kx if x is an element of degree [x[ of an A~-module M.

Fix an integer n E P' and denote by Pn : Z --* Z the multiplication by n. For a group K, the function (p=)* : Horn(Z, K) ~ Hom(Z, K) corresponds precisely to the nth power map z ~-, x = in K 22 under the obvious bijection Horn(Z, K) ~ g . Thus we have to prove that (p~)* is a bijection when K = EG for some G; in other words, that pn is an E-equivalence. 2) Z & EZ -% Zp, in which q denotes the natural transformation associated to E. Here r/cannot be identically zero, because there is a non-trivial homomorphism from Z to an E-local group-namely, Zp.