Unstable Modules Over the Steenrod Algebra, Functors, and the Cohomology of Spaces

Abstract

This report presents the connections discovered in the 80’s and the 90’s between homotopy theory, specifically unstable modules over the Steenrod algebra and homotopy classes of maps from classifying spaces, and certain categories of functors. These categories of functors are deeply linked to modular representation theory of symmetric or general linear groups.

In section 1 to section 4 we will present the topological background concerning modules over the Steenrod algebra. In section 5 to section 9 we will define and study the categories of functors in question and explain relations and applications. Here is the summary:

• 1. Unstable modules and algebras

• 2. Free objects in the category u

• 3. Injective objects and representability

• 4. Injectivity of the mod 2 cohomology of elementary abelian group and Lannes’ functor T_V

• 5. u/Nil and analytic functors

• 6. The functor ${p_n}:\mathcal{U}/\mathcal{N}il \to \mathcal{M}o{d_{{F_2}\left[ {{S_n}} \right]}}$, the filtration on u/Nil, and simple objects

• 7. The decomposition of the Carlsson modules K(i)

• 8. The Krull filtration on u

• 9.The Grothendieck ring of u