The principal investigator will attempt to broaden the scope of certain algebraic techniques of proven geometric utility. In the 1960's Daniel Quillen showed how to define homology and cohomology in many categories. These included simplicial sets, where he obtained the ordinary cohomology of topological spaces, but the categories of simplicial algebras over a field and simplicial unstable algebras over the mod-p Steenrod algebra both support an idea of cohomology that is relevant in homotopy theory. Cohomology of simplicial unstable algebras, for example, is the correct context for studying the E2 term of the Bousfield- Kan spectral sequence, an unstable Adams-type spectral sequence which has gained prominence due to the work of Haynes Miller, Jean Lannes and others. Under prior NSF grants, the principal investigator has made a successful study of the Quillen-type cohomology in various categories of algebras. The purpose of this project is to take these results and to extend and apply them. In particular, he should now be able to make an in-depth study of Barratt's desuspension spectral sequence and his "Lie ring analyzer" - both a source of conjecture and speculation for 30 years. Other projects include combining the Quillen-type cohomology techniques with the methods of Chris Stover to approach the homotopy groups of a wedge of two spaces, the study of a spectral sequence for computing the homotopy groups of a space of sections, and an exploration of the homology of function complexes using tools of Jean Lannes and A.K. Bousfield.
