VI INTRODUCTION We reprove the quasi-regularity theorem of [10] and the mod 3 decomposition of F., [ll] along with a new mod 5 decomposition of ER. The classification problem divides into two parts. The first part is to classify the possible rational cohomology rings and our methods say nothing about this. The second part is to construct (classify) for each prime p the mod p finite homotopy types which are H-spaces. In particular given a p-local H- space Y with mod p cohomology that of a finite complex, one can construct a finite H-space X such that Y is a factor of X, v, the p-localization of X. The methods of this paper interpret the result that a space is an H-space if and only if its k-invariants are primitive by providing a systematic frame- work in which to construct the p-local H-spaces, under certain assumptions on the mod p cohomology rings. The bulk of the paper is devoted to one such construction which we now de- scribe. Let T be the algebra over the mod p Steenrod algebra given by V A ( VVi ) 9 Z / p Z [ V 1 / ( x U ' where subscripts denote dimension and the Steenrod operations are specified by plX 3 = X 2p+1 6X 2p+l = X 2p+2 * THEOREM B. For each odd prime p there exists a simply connected finite CW complex K(p) whose localization at p is an H- space and whose mod p coho- mology is isomorphic to T as algebras over the Steenrod algebra. The proof of Theorem B is given in Chapter 3 after material of a general nature is developed in Chapter 2. We describe this now. For spaces Y with cohomology of a certain form, Massey and Peterson con- struct in [20] an unstable version of an Adams resolution of Y, [l]. The assumption on the mod p cohomology algebra of Y is that there exists M, an unstable module over the Steenrod algebra such that H*(X) = U(M), the free unstable algebra over the Steenrod algebra generated by M. The resolution of Y consists of a sequence of principal fibrations {E } Y - E s I E — K = g.e.m. s-i s & converging to Y in a suitable sense. The sequence of fibrations is constructed from a projective resolution of M. An important feature of these fibrations is a short exact sequence, called the fundamental sequence in [20],which gives a

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