Let LX be the space of free loops on a simply connected manifoldX. When the real cohomology of X is a tensor product of algebras generated by a single element, we determine the algebra structure of the real cohomology of LX by using the cyclic bar complex of the de Rham complex Ω(X) of X. In consequence, the algebra generators of the real cohomology of LX can be represented by differential form...

The syntomic cohomology, more precisely the cohomology of the sheaves s(n) on the syntomic site of a scheme, where introduced in [FM87] in order to prove comparison isomorphisms between crystalline and p-adic étale cohomology. It can be seen as an analogue of the Deligne-Beilinson cohomology in the p-adic world (for an excellent discussion see [Nek98]). In particular, when X is a smooth scheme ...

We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ = 0, and the algebraic operators D(p) are constructed from skew-products of p gamma matrices. We exhibit a large numb...

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves [V2], [HP1], [K4]. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the correspondin...

The present article continues the study of the boundary cohomology of Shimura varieties initiated in [HZ1, HZ2]. Let G be a reductive group over Q, X the symmetric space associated to G(R), and Γ an arithmetic subgroup (e.g., a congruence subgroup) of G(Q). We consider the cohomology of Γ\X with coefficients in the local system ̃ V constructed from a representation V of G, i.e., H•(Γ\X, ̃ V) ' H•...

A coring approach to non-Abelian descent cohomology of [P. Nuss & M. Wambst, Non-Abelian Hopf cohomology, Preprint arXiv:math.KT/0511712, (2005)] is described and a definition of a Galois cohomology for partial group actions is proposed.

We show that Haefliger's cohomology forétale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (oř Cech) cohomology for topological simplicial spaces.

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M . Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two application...