Variations on the cohomology of loop spaces on generalized homogeneous spaces

The cohomology of the space of loops on generalized homogeneous spaces is determined by using the Eilenberg-Moore spectral sequence. This generalizes classical results for homogeneous spaces of compact Lie groups.     

Stable cohomology of compact homogeneous spaces

The cohomology of certain compact homogeneous spaces is studied. The notion of stable cohomology (invariant under the passage to a finite covering) is introduced; examples of the calculation of this cohomology (Theorem 1) and its application to the study of the structure of compact homogeneous spaces (Theorem 2) are given. Several conjectures about properties of stable cohomology related to various areas of mathematics (such as topology and the cohomology of discrete (in particular, polycyclic) groups) are stated.     

On the cohomology of homogeneous spaces of finite loop spaces and the Eilenberg–Moore spectral sequence

We prove a collapse theorem for the Eilenberg–Moore spectral sequence and as an application we show that under certain conditions the cohomology of a homogeneous space of a connected finite loop space with a maximal rank torsion free subgroup is concentrated in even degrees and torsionfree, generalizing classical theorems for compact Lie groups of Borel and Bott.     

Higher cohomology of parabolic actions on certain homogeneous spaces

We show that for a parabolic R ^d -action on PSL(2,R) ^d /Γ, the cohomologies in degrees 1 through d ? 1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers, we have established these results for certain Anosov systems. This work extends the methods of those papers to systems that are not Anosov. The main new idea is defining special elements of representation spaces that allow us to modify the arguments from the previous papers. We discuss how to generalize this strategy to R ^d -systems coming from a product of Lie groups, as in the systems analyzed here.     

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,?)×…×SL(2,?)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Liv?ic’s theorem for Anosov flows for a standard partially hyperbolic ? ^d - or ? ^d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,?) ^d /Γ and are a step in the direction of the “top-degree cohomology” part.     

On the cohomology of stable map spaces

We describe an approach to calculating the cohomology rings of stable map spaces M̄_0,0( ⁿ ,d).     

On graded generalized local cohomology

Let be a homogeneous Noetherian ring with local base ring (R₀,m₀) and let M,N be two finitely generated graded R-modules. Let denote the i-th graded generalized local cohomology of N relative to M with support in . We study the vanishing, tameness and asymptotical stability of the homogeneous components of . Received: 22 March 2005; revised: 25 June 2005     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

On the signature of homogeneous spaces

We compute the signature of real and quaternionic Grassmannians, thereby completing the table of signatures of symmetric spaces given in a previous paper [4]. In addition, all homogeneous spaces of exceptional Lie groups with non-zero signature are listed.     

On Artinian generalized local cohomology modules

Let R be a commutative Noetherian ring with non-zero identity and a be a maximal ideal of R. An R-module M is called minimax if there is a finitely generated submodule N of M such that M/N is Artinian. Over a Gorenstein local ring R of finite Krull dimension, we proved that the Socle of H _a ⁿ (R) is a minimax R-module for each n ≥ 0.     

On the cohomology of free and twisted loop spaces

A natural extension of the cohomology suspension to a free loop space is constructed from the evaluation map and is shown to have good properties in cohomology calculation. This map is generalized to a twisted loop space which is a space of paths twisted by a given self-map of the underlying space. As an application, the cohomology of free and twisted loop spaces of classifying spaces of compact Lie groups, including some finite Chevalley groups, is calculated.     

On the theory of homogeneous Lipschitz spaces and mean oscillation spaces

In this paper the equivalence between the mean oscillation spaces and the homogeneous Lipschitz spaces will be shown through the use of elementary and constructive means. The mean oscillation spaces have been previously defined by Ricci and Taibleson for the case where the dimensionn=1. These spaces are extended here in a natural way to IR ⁿ .