viii

INTRODUCTION

In more detail here is a description of both the content and motivation for

the material in chapter 3.

Nov the calculation of objects Ext*(A,B) has a satisfactory algorithm in

terms of resolutions, and provides much opportunity for experimentation. For

the case at hand, this calculation was made in 197*+ and circulated in preprint

form. It is fair to say that this paper is a result of the reaction to that

preprint—congenial towards the result and hostile towards the argument. It

should be noted that 2.U.1 was not stated in the preprint but was implicit in

the argument.

Even with 2.U.1 explicit, the calculation itself will be found by many to be

indigestible. The version given here is to relate the calculation to well

known results concerning the stable cohomology of the Steenrod algebra. First

exact sequences and an algebraic EHP sequence are applied to the first variable

M. Next the spectral sequence obtained by filtering the second variable L is

used. The cohomology of the Steenrod algebra is used to calculate differ-

entials in the spectral sequence. There are quite a few, but each one is a

routine, and indeed trivial, calculation.

The final calculation is carried out in chapter 3, section k. Sections 2 and

3 are devoted to the construction of K(p) with appropriate self-maps so as to

apply 2.U.I. An analysis of the first two stages of its Adams resolution is

made to handle the situation at s = 1. The analysis results in an H-structure

on the first 3-stages of a resolution for K(p) and means that 2.^.1 can be

applied for s 2. Certain material preliminary to these calculations is

developed in chapter 1, sections 3, h. The material in section 3 is first used

in chapter 3, section 2 and that of section h is used only in the proof of

Proposition 3.^.2.

While Theorem 2.U.1 provides a systematic framework in which to construct

mod p H-spaces, its effectiveness is currently limited by the complexity of

calculations.

Finally I wish to thank many people who have offered criticisms and sugges-

tions. I'm indebted to Haynes Miller for much valuable advice concerning the

use of homological algebra. In fact his interest and concrete suggestions have

been a sustaining force during darker moments of this enterprise. I'm grateful

to A. Zabrodsky for his earliest versions of the lifting theorem. Thanks are

also extended to J. Stasheff for suggestions concerning the exposition, to

W. Browder and J. Lin whose work and comments provided much stimulating motiva-

tion and to A. Liulevicius who first indicated to me the scope of the ideas

used throughout this paper. I am also indebted to the topologists at North-

western and San Diego for making possible several visits to their institutions.

These visits had an indirect but valuable influence on my opinions.