INTRODUCTION vii tight grip on the cohomologies of the E . The fundamental sequence is an ex- tension of terms determined "by a resolution of M. By analogy, the fundamental sequence plays a role similar to that of the Serre exact sequence of a fibra- tion in the stable range. The relevant features of these fibrations are ex- tracted from [20] and presented in the definition of a Massey-Peterson fibra- tion in section 2 of chapter 1. The "well known use of Postnikov systems to study spaces can be mimicked in the use of Adams resolutions. In the stable range this has been standard procedure for years. Through the fundamental sequence, both structural and computational features of algebra can be brought to bear on problems which might be beyond reach in the ordinary Postnikov system. Chapter 2, section 1 consists of an exposition of unstable Adams resolutions, as developed in part III of [20], aimed toward their use in obstruction theory. Chapter 2, section 2 exploits the cohomological features of the resolution to obtain purely algebraic criteria for lifting maps through resolutions. This material is analogous to work of Toda [28] and Gitler and Mahowald [9] in stable homo- topy theory. In order to remove irrelevant algebraic considerations, recent results of Zabrodsky [30], [31] are combined with the material of sections 1 and 2. An exposition is given in section 3 including a simple form of Zabrodsky!s Lifting Theorem. All this material is applied in section k to a general result concerning the construction of mod p H-spaces. We describe this next. Given an H-space Y with cohomology U(M), a further necessary piece of structure comes from the power maps on Y. For each mod p integer e, the e-th power map J : Y -•Y has the property that the endomorphism $* of QH*(X) is multiplication by e. Turning the situation around, given a space Y with such a self-map \, a result of purely algebraic character is proved which states sufficient condi- tions for Y to be a mod p H-space. We review this. Let Horn ( , ) denote the degree preserving homomorphisms in the category of unstable modules over the Steenrod algebra, and Ext*( , ) the derived functors. Let L c H*(Y A Y: Z/pZ) be the submodule of e-characteristic vectors of (c| A j)*, L = {Z e H*(Y A Y) I (cj ) A |)*(z) = eZ} . THEOREM 2.U.I. If ExtS+1 (M,ISL) = 0 for all s 1, then Y is a mod. p E-epace (£ is suspension ). In chapter 3, the theory is applied to prove Theorem B. In this case T = U(M) with M = {x ,p2x ,BP2x } . Then it turns out that ExtS+1(M,ESL) = 0 for all s except s = 1. However the exceptional case is exactly where facts concerning the cohomology of 2-stage Postnikov systems can be applied.

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