Topology

Algebraic Topology Homotopy and Group Cohomology: by Alejandro Adem (auth.), Jaume Aguadé, Manuel Castellet,

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

The papers during this assortment, all totally refereed, unique papers, mirror many elements of contemporary major advances in homotopy thought and crew 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.- 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 solid splitting: the classifying house of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy area of expertise of BU(2) on the top 2.- G. Mislin: Cohomologically principal components and fusion in groups.

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Additional resources 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

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C h and Cg~ are homotopy equivalent. CII W c f l and Cg~ W Cgl are homotopy equivalent. C11 and Cg~ are not homotopy equivalent Since k and m are coprime, there are integers u and v with uk - vm = 1. = ® A ( : ) ' ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = H(gl V g~)(A(~) O h ) = H((gl v g~) o ¢) and Z(~ o (fl V f2)) = A(~)Z(fl V f2) 1 0 0 0 0 ky vy 0 0 k v 0 0 m u 0 0 0 0 1 0 0 0 0 0 0 0 uky 0 whereas Hence E(~o(fl V f2)) -~ E((gl Vg2)o¢) since u k - 1 (mod m), showing that CI1 VCi2 and Cg, V Cg2are homotopy equivalent, thereby completing the verification of (i).

K(P) is a spine of a closed prime orientable 3-manifold M(P) if and only if the permutation group generated by A ( P ) C ( P ) and B ( P ) C ( P ) (resp. A ( P ) and C ( P ) ) is transitive and the relation IA(P)I - IC(P)[ + 2 = IA(P)C(P)I holds. Here 101 denotes the number of cycles of a permutation 0 : E , E. 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.

Since every homomorphism ~: K - - * L with L P-local satisfies ~(Tp,(K)) = 1, the projection K - r , K / T p , ( K ) is a P-equivalence and hence a P-localization. It follows that the kernel of l: G ~ Gp is G n Tp,(K), which is contained in ETp,(G). 6 Let the infinite cyclic group C = (~) act on the abelian group A = Z[1/q] by ~. a = (1/q)a, for a certain positive integer q. Let S be the semidirect product A>4C with respect to this action. One can directly check that the nth power m a p is injective in S for all positive integers n, and hence Tp,(S) = 1 for every set of primes P.

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