INTRODUCTION The central goal of this paper is the construction of new finite H-spaces exhibiting torsion in homology. Before describing the results, it may be worthwhile to discuss their context. Because of the analogy with Lie groups, much of the research on finite H-spaces consists in proving facts known to hold for Lie groups. One of these facts, due to Borel [k] is that the integral homology of any compact simply connected Lie group is p-torsion free for primes p 7. Examples are constructed here which show Borel's result does not extend to simply connected finite H-spaces. THEOREM A. For each odd prime p, there exists a simply connected finite CW R-space x(.p) with nonzero -p-torsion in its integral homology. In partic- ular z/pZ is a summand of H2 +1 (x(p) Z). There are several consequences of Theorem A for the theory of finite H- spaces. These include showing that a number of results obtained by Browder [7]* Lin [l6] and Zabrodsky [29] are best possible. Roughly speaking the cohomological results allow for possibilities not found among Lie groups. Theorem A provides examples to fill the gaps. Another area to which Theorem A contributes is the problem concerning the relation of torsion and associativity. Emerging from the work of Zabrodsky [29] and Kane [13] is the fact that there is a basic difference, in the presence of torsion, between those homotopy types which admit homotopy associa- tive multiplications and those which do not. The exact relation of these properties is not yet clear. Theorem A is of some help towards perceiving the difference between homotopy associative multiplications and those just induc- ing associative homology rings. The first section of Chapter h enlarges on these applications of Theorem A and can be read independently from the rest of the paper. The results and methods of this paper are relevent to two other problems in the theory of finite H-spaces. These are mod p decomposition problems and the construction (classification) of finite CW complexes which admit multipli- cations. The former problem is discussed in sections 3 and ^ of Chapter k. Received by the editor August 2k 9 197b. Research supported in part by NSF Grant MCS-76-O7157. v

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