By Alejandro Adem (auth.), Jaume Aguadé, Manuel Castellet, Frederick Ronald Cohen (eds.)

The papers during this assortment, all absolutely refereed, unique papers, mirror many facets of modern major advances in homotopy idea and team cohomology. From the Contents: A. Adem: at the geometry and cohomology of finite uncomplicated 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.- ok. Ishiguro: Classifying areas of compact basic 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 distinctiveness of BU(2) on the best 2.- G. Mislin: Cohomologically primary parts and fusion in groups.

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**Additional info for Algebraic Topology Homotopy and Group Cohomology: Proceedings of the 1990 Barcelona Conference on Algebraic Topology, held in S. Feliu de Guíxols, Spain, June 6–12, 1990**

**Sample text**

1) I = To <_TI <_T2 <_... by letting Ti+l/Ti be the subgroup of G/TI generated by all its elements of type Tp,. Set Tx,,(G) = (jl T~. This subgroup of G was first analyzed by Ribenboim [20]. It has, among others, the following elementary properties. 1 (a) If G is P-local, then TF,(G) = 1. (b) Every homomorphism ~: G ---* If satisfies ~o(Tp,(G)) C_ Tp,(K). (c) For every group G, P'-roots are unique in G/Tp,(G). [] It follows from (a) and (b) that Tp,(G) is always contained in the kernel of l: G ---* Gp.

5 Remark: Provided it exists, the Afilm-localization of an object M is unique up to isomorphism and initial among maps from M to N//,,-closed objects. Hence the notation. 6 T h e o r e m . Afilm is a localizing subcategory of U. That is, for every object M of bl there exists its Afilm-localization. Proof'. 8] we only need to show that every object M of/4 has a largest m-nilpotent submodule. x is the sub-~L~-module of U generated by x. 7 Proposition. For every object M ofLt, M - ~ limA/',~'(M). t-----rn Proof:- Since N ' ~ I ( M ) is also N'it(m+l)-closed, we obtain the inverse system: --- --* A/'(I+D(M ) --* A/'r~l(M) -~ .

By selecting all possible cyclic orderings of the elements of Ei together their opposites in Ei, we may determine which permutations C ( P ) satisfy theorem 1. In this manner 52 we obtain a catalogue with possible repetitions (also using a computer) of all orientable prime closed 3-manifolds having K ( P ) as spine. As pointed out in [3] and [4], the permutations A ( P ) and C ( P ) allow us to draw a Heegaard diagram H ( P ) which represents M(P). Recall that a Heegaard splitting of a closed orientable 3-manifold M is a closed connected orientable surface Tg of genus g imbedded in M and dividing M into two homeomorphic handlebodies.