Realizing certain polynomial algebras as cohomology rings of spaces of finite type fibered over ×BU(d)

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The problem of constructing topologίcal 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 Oark 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 //-spaces, which 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 Oark 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 ^-completion of a simply connected CW complex of finite type. We will avoid these drawbacks by constructing for certain of the examples of Oark and Ewing a simply connected space of finite type with the requisite cohomology.