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

ISBN-10: 0387551956

ISBN-13: 9780387551951

The papers during this assortment, all absolutely refereed, unique papers, mirror many features of contemporary major advances in homotopy thought 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 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 good splitting: the classifying area of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy forte of BU(2) at the top 2.- G. Mislin: Cohomologically principal parts and fusion in teams.

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**Extra resources for Algebraic Topology: Homotopy and Group Cohomology : Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Held in S. Feliu De Guixols, **

**Example text**

Properties (1) and (2) ensure that there exists an elementary abelian p-group V and an embedding K C H ' V , [BZ]. Moreover if K satisfies (3) then there exists a subgroup G < G L ( V ) such that the N/10-1ocatization of K is the algebra of invariants H * V a, [BZ]. Having in mind the new construction of R m ( K ) and that T~vH*V ~- T ~ v ( O g * Y ) the proof goes like in the case p = 2 for m > 1. F. W. WILKERSON, Finite H-spaces and algebras over the Steenrod algebra, Ann. of Math. 111 (1980), 95-143.

S q k - l x = 0}, k > 1. 13 Example: If M E H is reduced then E k M is m-reduced for all m > k, but not (k - 1)-reduced because it is (k - 1)-nilpotent. 14 Lemma. If M is i-reduced then A f - l ( M ) is i-reduced for any s > O. Proo~ This is obvious for i _> s. Assume i < s. Then M has no s-nilpotent subA~-module and M ~ A/Z:(M) is injective; its cokernel, C, is s-nilpotent. Applying H i - : ( - ) to the obtained exact sequence, we get: M , A/~-: (M) t A/i-:(M) >C 1 , A/i-I(A/8-1(M)) t > 0 It follows that k e r ( A / Z i ( M ) --.

Proof'. We first observe that this inclusion is a Afil2,_i-isomorphism if and only if the inclusion in degree i: T ~ / ( ( K , (I)~H*VG)) C T~v ((If, q~aH*V) a) is an equality for any elementary abelian 2-group, W, and 0 < i < 2 ~ - 1. Next, the facts: (1) Tw(M) -~ (TwM) for any sub-A;-module M of an algebra H, and (2) (OaH*V)* = 0 if I < i < 2 ~ - 1, reduces the problem to prove that the inclusion:

### Algebraic Topology: Homotopy and Group Cohomology : Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Held in S. Feliu De Guixols, by Barcelona Conference on Algebraic Topology 1990 San Feliu De Guixols, Manuel Castellet, J. Aguade, Frederick R. Cohen

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