Pseudo-distances on symplectomorphism groups and applications to flux theory

Starting from a given norm on the vector space of exact 1-forms of a compact symplectic manifold, we produce pseudo-distances on its symplectomorphism group by generalizing an idea due to Banyaga. We prove that in some cases (which include Banyaga’s construction), their restriction to the Hamiltonian diffeomorphism group is equivalent to the distance induced by the initial norm on exact 1-forms. We also define genuine “distances to the Hamiltonian diffeomorphism group” which we use to derive several consequences, mainly in terms of flux groups.     

Topology of symplectomorphism groups of rational ruled surfaces

Let be either or the one point blow-up of . In both cases carries a family of symplectic forms , where -1$"> determines the cohomology class . This paper calculates the rational (co)homology of the group of symplectomorphisms of as well as the rational homotopy type of its classifying space . It turns out that each group contains a finite collection , of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups ``asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as . However, for each fixed there is essentially one nonvanishing product that gives rise to a ``jumping generator" in and to a single relation in the rational cohomology ring . An analog of this generator was also seen by Kronheimer in his study of families of symplectic forms on -manifolds using Seiberg-Witten theory. Our methods involve a close study of the space of -compatible almost complex structures on .

     

Cohomology of discontinuous groups

Prasad (1979) proved that the set of all equivalence classes of representationsp of a Fuchsian group Γ whose restrictions to the cyclic subgroups Γ _i -(c _i ) corresponding to the parabolic and elliptic elements of Γ occurring in the structure of Γ, are given, is a complex analytic manifold. In the process the author has proved thatH ¹(X,A)≈P ¹(Γ,ρ) and with suitable notation. This paper gives the corresponding results to the two above mentioned results, when in place of Γ we consider any discontinuous group of Poincare isometries Δ, and when similar assumptions are made.     

Cohomology and rigidity of fuchsian groups

We prove the localC ³-rigidity of the standard actions of cocompact lattices in PSL(2,ℝ) on a circle, using the Schwarzian and the duality technique for twisted cocycles. Partially supported by NSF Grant #DMS 9403870.     

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.     

Cohomology and finite subgroups of profinite groups

We prove two theorems linking the cohomology of a pro- group with the conjugacy classes of its finite subgroups.

The number of conjugacy classes of elementary abelian -subgroups of is finite if and only if the ring is finitely generated modulo nilpotent elements.

If the ring is finitely generated, then the number of conjugacy classes of finite subgroups of is finite.

     

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type and with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type with coefficients over the module Here the first standard generators of the group act by -multiplication, while the last one acts by -multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type as well as the cohomology of the classical braid group with coefficients in the -dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

     

Cohomology of topological groups and positive definite functions

In recent years, applications of cohomology theory to some aspects of probability theory have been found useful. In this paper, such applications are discussed for a determination of infinitely divisible positive definite functions on topological groups. This theory can be viewed as a new method for determining the class of such characteristic functions which includes the classical Leavy-Khintchine representation formula for the groups Rⁿ.     

Cohomology rings and formality properties of nilpotent groups

We analyze k-stage formality and relate resonance with this type of formality properties. For instance, we show that, for a finitely generated nilpotent group that is k-stage formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring, truncated up to degree k+1, of a finitely generated nilpotent, k-stage formal group is generated in degree 1; this criterion is necessary and sufficient for a finitely generated, 2-step nilpotent group to be k-stage formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.