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 points of modern major advances in homotopy idea and team cohomology. From the Contents: A. Adem: at the geometry and cohomology of finite basic 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 uncomplicated 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 good splitting: the classifying area of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy area of expertise of BU(2) at the top 2.- G. Mislin: Cohomologically relevant parts and fusion in teams.

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**Example text**

5 below. Its failure to be true for arbitrary groups is discussed in Section 4, cf. 7. One of the most characteristic features of generically trivial groups G is that homomorphisms ~: G ~ K are trivial for a broad class of groups K. 5 that this happens whenever K is residually nilpotent. 7 A group G is generically trivial if and only if every homomorphism ~: G --* K with K separable is trivial. PROOF. If G is generically trivial, then the composition r/ proj G ~ Ii ~ l l np---~ K p P is trivial for all p.

X, Y) of pointed maps from X to Y is weakly contractible. (c) EpX is contractible for every prime p and every idempotent functor E v in H o , satisfying Ep$1 ~_ ( $1)~. (d) Xp is contractible for all primes p. For the proof we need to remark the following fact. (X, Y) is weakly contractible. PROOF. Every map f : X --~ Y can be extended to the cone of X by obstruction theory, because f. : 7rl(X) --* 7rl(Y) is trivial and the cohomology groups of X with untwisted coefficients are zero. (X, Y) is path-connected.

Let P be any set of primes such that 2, 3 ¢ P, and G ~ K be a P-faithful embedding in which b has a cube root d. 1). Then one rea~lily checks that (a-lda) s = a~, and hence a-ldad -2 E T~; further (da-~)2c -2 E T~, which implies d(ca) -~ E T2, and so (ca)-Zb E T2. This element belongs to ETp,(G) and is not trivial. We can now give the following characterization of generically trivial groups. 7 A group G is generically trivial if and only if, for every prime p, there exists a p-faithful embedding G '--* K such that G C Tp,(K).