There is a relation between the chromatic filtration of stable homotopy theory and a stratification of the algebraic variety associated to the cohomology of a finite group which was made explicit by the work of Green, Hunton and Schuster [4]. The purpose of this paper is to investigate generalizations of this theory, by exploiting the theory of unstable algebras over the Steenrod algebra for si...

ON THE HOPF CONJECTURES WITH SYMMETRY Lee Kennard Wolfgang Ziller, Advisor We prove results related to the classical Hopf conjectures about positively curved manifolds under the assumption of a logarithmic symmetry rank bound. The main new tool is the action of the Steenrod algebra on cohomology. Along the way, we formulate a certain notion of periodicity in cohomology and obtain a generalizati...

We characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition. When the mod p cohomology of an H-space is finitely generated as an algebra over the Steenrod algebra we prove that its homotopy groups behave like those of a finite complex. In particular, a p-complete infinite loop space has a finite number o...

Suppose that F is a number field, with ring of integers OF . Let ` denote an odd prime and let R = OF [1/`]. In [3], the author and W.Dwyer gave an explicit conjectural computation of the mod ` cohomology of the infinite general linear group GLR. Here is the quickest and simplest statement of the conjecture (all homology and cohomology groups have Z/` coefficients): let U denote the infinite un...

Let X be a 1-connected compact space such that the algebra H(X;F2) is generated by one single element. We compute the cohomology of the free loop space H(ΛX;F2) including the Steenrod algebra action. When X is a projective space CP, HP, the Cayley projective plane CaP or a sphere Sm we obtain a splitting result for integral and mod two cohomology of the suspension spectrum Σ(ΛX)+. The splitting...

We investigate support schemes for infinitesimal unipotent supergroups and their representations. Our main results provide a non-cohomological description of these which generalizes the classical work Suslin, Friedlander, Bendel. As consequence, in this setting have desired features such theory, including naturality with respect to group homomorphisms, tensor product property, realizability. an...