Haefliger 可微上同调

Differentiable Cohomology of Gauge Groups

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilinson lead to differentiable cohomology classes with coefficients in C*. These may be viewed as an enrichment of the Chern-Simons differential forms. By transgression, classes in differentiable cohomology of a Lie group G lead to differentiable cohomology classes for gauge groups Ma…

Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry

We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic structure. We establish a canonical isomorphism between the periodic Hopf cyclic cohomology of this extended Hopf algebra and the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then show that this isomorphism can be explicitly implemented at the cochain …

Leibniz Cohomology and Connections on Differentiable Manifolds

We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Additionally, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean ${\bf{R}}^n$ corre…

Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu). As a second application we extend van Est's argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives (a slight improvement …

关于可微栈的上同调

On the cohomology of differentiable stacks

本文证明李群胚以同伦意义下的表示为系数的上同调是Morita不变量，从而成为可微栈的不变量，并附带证明高阶李群胚的上同调下降性质。

关于李群胚的Bott谱序列的若干观点

Perspectives on the Bott spectral sequence for Lie groupoids

本文关联统一两类改进可微栈上同调计算的方法：Arias Abad将Bott谱序列推广至李群胚，作者则提出正则真李群胚对应可微栈的上同调模型。

On the de Rham Cohomology of Differential and Algebraic Stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack …

Differentiable Categories, gerbes and G-structures

The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like orbifolds are not often smooth. In classical differential geometry, non smooth spaces appear also naturally, for example in the theory of foliations, the space of leaves can be an orbifold with singularities. The scheme to study these structures is identical: classical tools used in differential geometry, like connections, curvature are adapted. The purpose …

微分几何中的德雷哈理论

De Rham Theory in Derived Differential Geometry

本文研究了奇异微分流形的德雷哈理论，提出两种可能的答案，并展示了在特定条件下，德雷哈上同调与常sheaf上同调之间的关系，以及如何通过条件使局部不变量消失，从而得到德雷哈型同构。

Differential characters as stacks and prequantization

We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes.

Three Cocycles on $\Diff(S^1)$ Generalizing the Schwarzian Derivative

The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the Möbius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules of linear differential operators on $S^1$ is calculated. We introduce three non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the Schwarzian derivative.

The van Est Map on Geometric Stacks

We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies differentiable stacks, Lie algebroids and homotopy theory. In Part 2 of this thesis we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the cohomology of a simple foliation $\mathcal{H}\to\mathcal{G}$. In Part 1 we generalize the functions that we can take cohomology of…