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