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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, mirror many points of modern major advances in homotopy conception 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 strong splitting: the classifying area of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy strong point of BU(2) at the top 2.- G. Mislin: Cohomologically important parts and fusion in teams.

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Moreover, we treat E ( f ) as a column Kvector over T, the torsion subgroup of 7r4n_l(S2n). (cf. [1]). Recall further that homotopy (resp. /)-)equivalences correspond to diagrams where both det(A_A_(qa)) and det(A_A_(~b)) = :hl (resp. both coprime to p). We restrict attention to n ¢ 1, 2, 4 merely to avoid the minor technical inconviences occasioned by the existence of maps Hopf invariant 1. Take y E T of order m. Take 1 < k < m - 1 with k co-prime to m. (Note that this is possible if and only if m ¢ 1,2, 3, 4, 6) Choose fi, gi : S 4~-1 ) S 2n V S 2" given by 0 001 We (i) (ii) (iii) (iv) [000 claim that: Cfl V Cf~ and Cg 1 V Cg2 are homotopy equivalent.

Also, via its regular representation, a finite p-group of order n embeds in the group of n x n unitriangular matrices over Fp. Thus every finitely generated nilpotent group embeds in K. By the above remark, such an embedding must be P-faithful. Thus K is a countable, acyclic group such that, for all sets of primes P, Kp contains a copy of Gp for every finitely generated nilpotent group G. 27 A group is called P'-radicable [23] if each of its elements has at least one n t h root for every n E P ' .

Interesting examples of this kind are - the infinite alternating group Aoo; - Philip Hall's countable universal locally finite group [14]; - all non-trivial algebraically closed groups [17]. 6 Every infinite group G can be embedded in a generically trivial group of the same cardinality as G. o By [6], every abelian group is the centre of a generically trivial group. This prompts the question: which groups are normal in generically trivial groups? 7 For any sequen~ce A2, A3, . . , A~, . . o f abelian groups, there exists a generically trivial group G such that Hn(G; Z) ~- A,~ f o r all n ~ 2.

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