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

The papers during this assortment, all totally 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 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 reliable splitting: the classifying house of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy strong point of BU(2) on the leading 2.- G. Mislin: Cohomologically valuable 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**

**Sample text**

Let B ( P ) be the involutory permutation of E in which e hi and ~h are corresponding points. An arbitrary numbering of the elements of E around each vertex ei (resp. ~i) determines a permutation C ( P ) of E whose orbit sets are Ei, Ei. With the above notation, we have the following result proved in [14]. T H E O R E M 1. 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.

It is easy to show that the algebra of invariants Rs(K) c is now the integral closure of A/~-X(K) in its field of fractions. As a consequence, if K in the corollary above is integrally closed we also have K = Rm(K) a. Proof of theorem: Let K be a connected unstable A~-algebra. Properties (1) and (2) ensure that there exists an elementary abelian 2-group V with dimension the krulldimension of K and an embedding K C H ' V , [AW]. of K is the algebra of invariants: H*V c ([AW] [BZ]). C~m+lH*V) G i2 , Af~I(K ) ) Rm(K) c The inclusions K C

17 P r o p o s i t i o n . /~fil m-clOsed, and N is (exactly) j-reduced and A/iln-closed. Then, M ® N is (exactly) (i + j)-reduced and A/ilm~xl( i+,),(j+m) } -closed. " We already know M ® N is (exactly) (i + j)-reduced. Suppose (n + i) _> (m + j). Let N --* A/j-I(N) ~ C be the A/i/j-localization of N. Then, for k > j, ~kC ~-- (R~Ek)(N), so 'C is (exactly) n-reduced. Now we consider the exact sequence 0 ~ M ® N ~ M ® A/TI(N) ~ M ® C ~ 0. )-closed. 18 P r o p o s i t i o n . Every projective limit of A/ilm-closed objects is A/ilm-elosed.