CHAPTER 1. ALGEBRAIC INFORMATION

0. INTRODUCTION. The natural setting for the algebraic parts of our

work is in the category of unstable modules over the Steenrod algebra intro-

duced in [27]. In §1.1 we review those portions of the theory we require.

References are [5],[6], [20],[26], [27]. There are 15 items. Many of them

concern the algebraic loop functor ft, which plays a central role.

In §1.2 we discuss the notion of a Massey-Peterson fibrat'lon, These were

introduced in [20] for the prime 2 and the basic theory for odd primes was

proved by Barcus in [3]. After §1.2,the reader may prefer to skip the re-

maining sections of this chapter and go to chapter 2 where the basic theory of

unstable Adams resolutions is developed.

Sections 3 and h are concerned with some special facts from homological

algebra. These facts are needed for the calculation of certain unstable Ext

groups which arise in the proof of Theorem B. The general theory leading to

this type of calculation is given in chapter 2, §2. The material of §1.3 is

first used in §3.2 and that of §1.^ is used only in the proof of Proposition

3.U.2.

1. ALGEBRAIC INFORMATION

Here material needed for the discussion is assembled.

1.1. RECOLLECTIONS AND NOTATIONS

References are [5],16],[20],I26], I27].

1.1.1. The mod p Steenrod algebra is denoted by Q and only odd primes

are considered. Let B(.n)c G be the1 left ideal of operations an-

nihilating classes of dimension n. An Q-module M is unstable

provided [27]

B(n)Mn =0 for all n 0 .

We write Uft/G for "the category of unstable left G-modules and de-

gree preserving homomorphisms. An algebra W over Q is unstable

provided it is an unstable module and

p x = x , dim x = 2n .

1.1.2. STEENRODrS U FUNCTOR [27]. Let M be a connected unstable G-

module. We write U(M) for the free G-algebra generated by M

constructed in [27]. The canonical embedding is written

j: M - U(M). The functor U is left adjoint to the forgetful

functor from unstable algebras to unstable modules.

1