Cobordism for Hamiltonian loop group actions and flat connections on the punctured two-sphere

We develop a theory of symplectic cobordism and a Duistermaat-Heckman principle for Hamiltonian loop group actions. As an application, we construct a symplectic cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties. This cobordism yields formulas for the mixed Pontrjagin numbers of the moduli spaces, equivalent to Witten's formulas in the case of symplectic volumes. Received June 15, 1998     

Witten's nonabelian localization for noncompact Hamiltonian spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303-368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.     

Small loop spaces

The importance of small loops in the covering space theory was pointed out by Brodskiy, Dydak, Labuz, and Mitra in [2] and [3]. A small loop is a loop which is homotopic to a loop contained in an arbitrarily small neighborhood of its base point and a small loop space is a topological space in which every loop is small. Small loops are the strongest obstruction to semi-locally simply connectedness. We construct a small loop space using the Harmonic Archipelago. Furthermore, we define the small loop group of a space and study its impact on covering spaces, in particular its contribution to the fundamental group of the universal covering space.     

The finiteness obstruction for loop spaces

For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite CW-complex or not. It was conjectured that this finiteness obstruction always vanishes for quasi finite H-spaces, that are H-spaces whose homology looks like the homology of a finite CW-complex. In this paper we prove this conjecture for loop spaces. In particular, this shows that every quasi finite loop space is actually homotopy equivalent to a finite CW-complex. Received: March 25, 1999.     

On loop spaces of configuration spaces

This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as ``twistings" between the factors. The main homological results are given in terms of extensions of the ``infinitesimal braid relations" or ``universal Yang-Baxter Lie relations".

     

Twistor spaces and spinors over loop spaces

In this paper two natural twistor spaces over the loop space of a Riemannian manifold are constructed and their equivalence is shown in the Kählerian case. This relies on a detailed study of frame bundles of loop spaces on the one hand and, on the other hand, on an explicit local trivialization of the Atiyah operator family [defined in Atiyah (SMF 131:43–59, 1985)] associated to a loop space. We relate these constructions to the Dixmier-Douady obstruction class against the existence of a string structure, as well as to pseudo - line bundle gerbes in the sense of Brylinski (Loop spaces, characteristic classes and geometric quantization. Birkhäuser, Basel, 1993).     

Surjectivity of differential operators and the division problem in certain function and distribution spaces

We give an overview on surjectivity conditions for partial differential operators and operators defined by multiplication with polynomials on certain function and distribution spaces of Laurent Schwartz. We complement the classical results by treating the surjectivity of operators on the space of slowly increasing functions and on the space of rapidly decreasing distributions, respectively.     

非光滑映射的局部满射定理

本文首先将Ivar Ekeland新近证明的一个反函数定理,推广到非光滑映射的情形.以此结果为基础结合线性算子广义逆的稳定性理论给出B.H.Pourciauhas关于非光滑映射的局部满射定理的一个新的证明,新证明方法较原先作者的方法更简洁.     

Configuration spaces on the sphere and higher loop spaces

We show that the homology over a field of the space ⁿⁿX of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space C(Sⁿ,X) on the n-sphere with labels in X and of its completion, that depend only on the homology of X. In many but not all cases the homology of C(Sⁿ,X) coincides with the homology of ⁿⁿX. In particular we obtain the homology of the unordered configuration spaces on a sphere.Mathematics Subject Classification (2000): 55P48, 55R80, 55S12in final form: 30 December 2003     

Characteristic classes and obstruction theory for infinite loop spaces

The classical extension problem is to determine whether or not a given mapg:AY, defined on a given subspaceA of a spaceX, has an extensionXY. The present paper examines this question in the special case where thek-invariants ofY are cohomology classes of finite order (for instance ifY is an infinite loop space).     

Homotopy nilpotency in p-regular loop spaces

We consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p-completion of a product of finite numbers of spheres. We determine the homotopy nilpotency of those loop spaces as an answer to this problem.     

Differential calculus on path and loop spaces II. Irreducibility of Dirichlet forms on loop spaces

We prove the irreducibility of a Dirichlet form on the based loop space on a compact Riemannian manifold. The Dirichlet form is defined by the gradient operator due to Driver and Léandre. We also prove the uniqueness of the ground states of the Schrödinger operator for which the Dirichlet form satisfies the logarithmic Sobolev inequality. This is an extension of the corresponding results of Gross ([28], [29]) to the case of general compact Riemannian manifolds.