In Part I of the thesis I investigate the invariants of the divided powers algebra, Γ(<i>V</i>) under the action of the General and Special Linear groups. In Chapter 2 I compare the additive structure of the invariants of Γ(<i>V</i>) with that of the invariants of the polynomial algebra. I show that these are not isomorphic as vector spaces (and <i>a fortiori</i> not isomorphic as algebras) except in two cases – when dim(<i>V</i>) = 2 and <i>p</i> = 2 or 3. This chapter also includes some dimension counting arguments most notably the case dim(<i>V</i>) = 2, <i>p</i> = 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(<i>V</i>) = 2 and <i>p</i> = 2 or 3 in sections 3.4 and 3.6 respectively. In addition I describe the algebra structure of the invariants of Γ(<i>V</i>) under some important subgroups of <i>GL</i>(<i>V</i>) – the transvections, the symmetric subgroup and the multiplicative subgroup. I give complete results for the transvections and F<i><sub>p</sub></i><sup>x</sup><sub>.</sub>. For the symmetric invariants I correct a result of Joel Segal and give a complete description of Γ<i><sub>p</sub></i>(<i>V<sub>p</sub></i>)<sup>Σ<i>p</i></sup><sub>.</sub>. The method used in the description of Γ(<i>V</i>)<i><sup>SL</sup></i><sup>3(<i>V</i>2) </sup>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 K-theory squares are defined on any space homotopic to a finite CW-complex with no 2-torsion. 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:439639 |
| Date | January 2006 |
| Creators | Salisbury, David S. |
| Publisher | University of Aberdeen |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0207 seconds