Title:

Results concerning the Steenrod algebra

In Part I of the thesis I investigate the invariants of the divided powers algebra, Γ(V) under the action of the General and Special Linear groups. In Chapter 2 I compare the additive structure of the invariants of Γ(V) with that of the invariants of the polynomial algebra. I show that these are not isomorphic as vector spaces (and a fortiori not isomorphic as algebras) except in two cases – when dim(V) = 2 and p = 2 or 3. This chapter also includes some dimension counting arguments most notably the case dim(V) = 2, p = 2 in Section 2.7. These dimension counts are useful both as techniques in their own right and because they give explicit calculations which are useful in the following chapter. In Chapter 3 I describe the algebra structure of the invariant algebra in the two cases, dim(V) = 2 and p = 2 or 3 in sections 3.4 and 3.6 respectively. In addition I describe the algebra structure of the invariants of Γ(V) under some important subgroups of GL(V) – the transvections, the symmetric subgroup and the multiplicative subgroup. I give complete results for the transvections and F_{p}^{x}_{.}. For the symmetric invariants I correct a result of Joel Segal and give a complete description of Γ_{p}(V_{p})^{Σp}_{.}. The method used in the description of Γ(V)^{SL}^{3(V2) }has potential to be extended to other cases. In Part II of the thesis I compare two different methods of defining some kind of Steenrod Operations in integral cohomology. John Hubbuck’s Ktheory squares are defined on any space homotopic to a finite CWcomplex with no 2torsion. Reg Wood’s differential operator squares are defined only on polynomial algebras. The question is whether considered on a suitable space these operations would be equivalent. I show that they are essentially incompatible.
