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
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
* Lin [l6] and Zabrodsky  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
 and Kane  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
Research supported in part by NSF Grant MCS-76-O7157.