Title:

Results concerning the Steenrod algebra

In this thesis I provide two main results concerning the Steenrod algebra, one relating to the mod 2 case and the other holding for all odd primes. In chapter 1 certain properties of the Adams operators on complex K theory, which were described by Adams [1] and Atiyah [3], are used to give a construction of the Steenrod squares on the cohomology of finite 2torsion free CW complexes. With this approach, certain periodic relations are clearly seen to hold in A(2), the mod 2 Steenrod algebra, which are not easily derived algebraically from the standard formulation of the Adem relations. It turns out that these formulae have been established previously in the literature (see [6], [21]) but the method of proof presented here is completely different. A variety of applications are considered, serving as examples of how these equations may be of use. Of particular interest is the observation that the ideal generated by these relations is a Hopf ideal and we use this to construct a Hopf superalgebra of A(2). Chapter 2 tackles the problem of calculating which classes in the homology of a kfold product of infinite complex projective spaces are annihilated by the dualised action of the elements of positive grading in A(p), the mod p Steenrod algebra. The space of all such classes is denoted by M_{k}. Our objective in calculating M_{k} is to enumerate all the irreducible representations of GL(k,F_{p}) over F_{p}. For k = 1 the problem is trivial and the result is mentioned briefly. The case where k = 2 forms the bulk of the chapter and the solution is computed directly. Some comments are made about the general situation, where k > 2, which may assist an attempt at solving this problem. M_{1} can be 'embedded' into M_{k} (for k > 1) and we denote by L_{k} the subalgebra generated by elements in the images of all such embeddings.
