Introduction to the Tangent Grupoid

We present some plausible definitions for the tangent grupoid of a manifold M, as well as some of the known applications of the structure. This is a kind of introductory note.Comment: 8 pages, late

Torsions for manifolds with boundary and glueing formulas

We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle \Cal F of \Cal A-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$ into disjoint components. In particular we extend the $L-2$ analytic and Reidemeister torsions to compact manifolds with boundary. If the system (M,\partial_-M, \partial_+M, \Cal F) is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove glueing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]Comment: 58 pages, amste

Groups quasi-isometric to symmetric spaces

We determine the structure of finitely generated groups which are quasi-isometric to symmetric spaces of noncompact type, allowing Euclidean de Rham factors If $X$ is a symmetric space of noncompact type with no Euclidean de Rham factor, and \Ga is a finitely generated group quasi-isometric to the product \E^k\times X, then there is an exact sequence 1\ra H\ra\Ga\ra L\ra 1 where $H$ contains a finite index copy of $\Z^k$ and $L$ is a uniform lattice in the isometry group of $X$.Comment: 10 pages, Late

Equivariant torsion of locally symmetric spaces

The equivariant holomorphic torsion of a compact locally symmetric manifold and an automorphism is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.Comment: LATEX, 24 page

The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

Let $G$ be a simple complex Lie group, \alg{g} be its Lie algebra, $K$ be a maximal compact form of $G$ and \alg{k} be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of \alg{g} which singles out the compact form \alg{k}. Consider the space of flat \alg{g}-valued connections on a Riemann sphere with three holes which satisfy the additional condition $\overline{A(z)}=-A(\overline{z})$. We call the quotient of this space over the action of the gauge group $\overline{g(z)}=g^{-1}(\overline{z})$ a \emph{hyperbolic} moduli space of flat connections. We prove that the following three symplectic spaces are isomorphic: 1. The hyperbolic moduli space of flat connections. 2. The symplectic multiplicity space obtained as symplectic quotient of the triple product of co-adjoint orbits of $K$. 3. The Poisson-Lie multiplicity space equal to the Poisson quotient of the triple product of dressing orbits of $K$.Comment: 19 pages, AMS LaTe

Twistor spaces of non-flat Bochner-K\"ahler manifolds

For the twistor spaces of the Bochner-K\"ahler manifold $M = H^l \times P^n$, systems of holomorphic coordinates are constructed. As an application of them, an explicit description of the moduli space of relative deformations of fibers of $M$'s twistor space is given.Comment: AMS-LaTeX v1.2, 11 page

New examples of conservative systems on S^2 possessing an integral cubic in momenta

It has been proved that on 2-dimensional orientable compact manifolds of genus $g>1$ there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on $T^2$ possess an integral quadratic in momenta. All geodesic flows on $S^2$ and $T^2$ possessing integrals linear and quadratic in momenta have been described by Kolokol'tsov, Babenko and Nekhoroshev. So far there has been known only one example of conservative system on $S^2$ possessing an integral cubic in momenta: the case of Goryachev-Chaplygin in the dynamics of a rigid body. The aim of this paper is to propose a new one-parameter family of examples of complete integrable conservative systems on $S^2$ possessing an integral cubic in momenta. We show that our family does not include the case of Goryachev-Chaplygin.Comment: 10 pages, AMS-LaTe

The singularities of Yang-Mills connections for bundles on a surface. I. The local model

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, not necessarily connected, with Lie algebra $g$, endowed with an adjoint action invariant scalar product, let $\xi \colon P \to \Sigma$ be a principal $G$-bundle, and pick a Riemannian metric and orientation on $\Sigma$, so that the corresponding Yang-Mills equations $$d_A*K_A = 0$$ are defined, where $K_A$ refers to the curvature of a connection $A$. For every central Yang-Mills connection $A$, the data induce a structure of unitary representation of the stabilizer $Z_A$ on the first cohomology group \roman H^1_A(\Sigma,\roman{ad}(\xi)) with coefficients in the adjoint bundle \roman{ad}(\xi), with reference to $A$, with momentum mapping $\Theta_A$ from \roman H^1_A(\Sigma,\roman{ad}(\xi)) to the dual $z^*_A$ of the Lie algebra $z_A$ of $Z_A$. We show that, for every central Yang-Mills connection $A$, a suitable Kuranishi map identifies a neighborhood of zero in the Marsden-Weinstein reduced space \roman H_A for $\Theta_A$ with a neighborhood of the point $[A]$ in the moduli space of central Yang-Mills connections on $\xi$.Comment: 14 page

On Rumin's Complex and Adiabatic Limits

This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the $L^2$-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory spaces seen from within '', IHES preprint, 1994. This result can also be reformulated in terms of spectral sequences, after Forman, Mazzeo-Melrose. A key ingredient in the proof is the fact that the curvatures become unbounded in a controlled way.Comment: 18 page

Extended moduli spaces and the Kan construction.II.Lattice gauge theory

Let $Y$ be a CW-complex with a single 0-cell, $K$ its Kan group, a model for the loop space of $Y$, and let $G$ be a compact, connected Lie group. We give an explicit finite dimensional construction of generators of the equivariant cohomology of the geometric realization of the cosimplicial manifold \roman{Hom}(K,G) and hence of the space \roman{Map}^o(Y,BG) of based maps from $Y$ to the classifying space $BG$. For a smooth manifold $Y$, this may be viewed as a rigorous approach to lattice gauge theory, and we show that it then yields, (i) when {\roman{dim}(Y)=2,} equivariant de Rham representatives of generators of the equivariant cohomology of twisted representation spaces of the fundamental group of a closed surface including generators for moduli spaces of semi stable holomorphic vector bundles on complex curves so that, in particular, the known structure of a stratified symplectic space results; (ii) when {\roman{dim}(Y)=3,} equivariant cohomology generators including the Chern-Simons function; (iii) when {\roman{dim}(Y) = 4,} the generators of the relevant equivariant cohomology from which for example Donaldson polynomials are obtained by evaluation against suitable fundamental classes corresponding to moduli spaces of ASD connections.Comment: AMSTeX 2.1, 21 page