The problem of constructing
topological spaces whose cohomology ring with coefficients in the field of p elements
is a polynomial algebra has attracted the attention of algebraic topologists for many
decades. Apart from the naturally occurring examples, classifying spaces of Lie
groups away from their torsion primes, rather little progress was made until the
construction of Clark and Ewing of a vast number of new non-modular examples.
The completeness of their construction in the non-modular case was shown by Adams
and Wilkerson (see Smith and Switzer for a compact-proof). One interest in the
construction of spaces with polynomial cohomology is that they are related
to the study of finite H-spaces, which should appear as their loop spaces;
“should” because the construction of Clark and Ewing does not yield a simply
connected CW complex of finite type. On the contrary the construction of
Clark and Ewing yields non-simply connected spaces that are p-adically
complete. By forming their finite completion they can be made simply connected.
But considerably more effort would be required to show that they have the
homotopy type of the p-completion of a simply connected CW complex of finite
type.
We will avoid these drawbacks by constructing for certain of the examples of
Clark and Ewing a simply connected space of finite type with the requisite
cohomology.
