Title:

Topological methods in algebraic geometry : cohomology rings, algebraic cobordism and higher Chow groups

This thesis is divided into three chapters. The first chapter is about the cohomology ring of the space of rotational functions. In the second chapter, we define algebraic cobordism of classifying spaces, Ω*(BG) and Gequivariant algebraic cobordism Ω*_{G}() for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^{j}(Ω*()); which are required for the definition to work. We show that Gequivariant cobordism satisfies the localization exact sequence. We compute Ω*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n + 1). We also compute Ω*(BG) when G is a finite abelian group. A finite nonabelian group for which we compute Ω*(BG) is the quaternion group of order 8. In all the above cases we check that Ω*(BG) is isomorphic to MU*(BG). The third chapter is workinprogress on Steenrod operations on higher Chow groups. Voevodsky has defined motivic Steenrod operations on motivic cohomology and used them in his proof of the Minor Conjecture.
