# Algebraic Topology, Poznan 1989: Proceedings of a Conference by S. Jackowski, B. Oliver, K. Pawaloski

By S. Jackowski, B. Oliver, K. Pawaloski

As a part of the clinical job in reference to the seventieth birthday of the Adam Mickiewicz college in Poznan, a world convention on algebraic topology was once held. within the ensuing court cases quantity, the emphasis is on gigantic survey papers, a few offered on the convention, a few written for that reason.

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One can then attach infinitely many free G-n-celis to M(fl) to obtain a G-complex W which has the "right" homology and turns out to be finitely G-dominated. Its equivariant finiteness obstruction wG(W ) corresponds to the projective class group invariant (-1)n[ Hn(M(fl) ) ] above. Results similar to those of Oliver and Petrie was also obtained (in less generality) by Ku and Ku [29]. In particular they have proved the following generalization of Swan's theorem. 3 ) Let G be a finite group of order q with periodic cohomology o f period n and let d = (q, ¢(q)) where ~b is the Euler C-function.

The following theorem is proved in §5. THEOREM D. space. having a finite Postnikov tower of fi~fite type localized at some prime p, then it follows that all the homolog3" groups of A- ~'ith coett~cients hi Zp in positive dimensions vanish. In particular, if X is a p local C W complex, it is contractible. This theorem D is derived from theorem A. THEOREM A. pace X with all the rood p homology groups finite di: menslonM has the property that the depth of the algebra H . ( ~ X , lp) is not bigger than the Lusternik-Schnilvhnaml category of X .

H. Dovermann: personal communication. 17. tL Dovermann, M. Rothenberg: An equivariant surgery sequence and equivariant diffeomorphlsm and homeomorphism classification, Topology Symposium (Siegen 1979), pp. 257-280, Lecture Notes in Math. 788, Springer Vlg 1980. 36 18. H. Dovermann, M. Rothenberg: Equivariant surgery and classification of finite group actions on manifolds, Memoirs Amer. Math. Soc. 379 (1988). 19. S. Ferry: A slmple-homotopy approach to the finiteness obstruction, Shape Theory and Geometric Topology, pp.