The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification of the boundary with a product M1 x P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S^1, we show how to attach to such a space a noncommutative C*-algebra that captures the extra structure. We then use this C*-algebra to give a new proof of the Freed-Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S^1 singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.
We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to lift singularities, in particular when one imposes a compatibility condition with an additional structure given by a multi-vector field.
We give a superconnection proof of Connes' index theorem for proper cocompact actions of etale groupoids. This includes Connes' general foliation index theorem for foliations with Hausdorff holonomy groupoid.
We show that there is an one-to-one correspondence between resolutions (equivariant w.r.t. a Lie groupoid action) of a singular subset of a manifold, and substacks (of a certain type) of the differential stack associated to the Lie groupoid in question. In particular, we show how to build an equivariant resolution out of Lie subgroupoids (of a certain type).
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.
We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the "topological side" of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.
In this paper we consider a family of Dirac-type operators on fibration $P \to B$ equivariant with respect to an action of an etale groupoid. Such a family defines an element in the bivariant $K$ theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map $Φ$ in the cyclic cohomology. A particular case of this result is Connes' index theorem for etale groupoids in the case of fibrations.
We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition as well as a computation of the "global …
We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the …
We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions of $C^*$-algebras and to full index problems. We compute the corresponding K-theory maps. Finally, the blowup of a manifold sitting in a transverse way in the space of objects of a Lie groupoid leads to a calculus, quite similar to the Boutet de Monvel calculus for manifolds with boundary.
For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in \cite{Ca2}. This allows in particular to prove a more primitive version of the Connes-Skandalis Longitudinal index Theorem for foliations, that is, an index theorem taking values in a group which pairs with Cyclic cocycles. As other application, for $D$ a $\gr-$PDO ellipt…
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside…
This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This groupoid has a natural geometric meaning as a holonomy groupoid of singular foliation, it is a singular leaf space in the sense of Androulidakis and Skandalis. We then show that the associated compactly supported pseudodifferential calculus coincides with Mazzeo and Melrose's phi-cal…
We construct the holonomy groupoid of any singular foliation. In the regular case this groupoid coincides with the usual holonomy groupoid of Winkelnkemper (1983); the same holds in the singular cases of Bigonnet and Pradines (1985) and Debord (2001), which from our point of view can be thought of as being "almost regular". In the general case, the holonomy groupoid can be quite an ill behaved geometric object. On the other hand it often has a nice longitudinal smooth structure. Nonetheless, we use this groupoid to generalize to the singular case Connes' construction of the C*-algebra of the f…
We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a furthe…
Let $M$ be a spin manifold, the Dirac operator with coefficient in the universal flat Hilbert $C^\ast π_1(M)$-module determines a "Rosenberg index element" which, according to B.Hanke and T.Schick, subsumes the enlargeablility obstruction of positive scalar curvature on $M$. In this note, we generalize this result to the case of spin foliation. More precisely, given a foliation $(M,F)$ with $F$ spin, we shall define a foliation version of "Rosenberg index element" and prove that it is nonzero at the presence of compactly enlargeability of $(M,F)$.
Let $Γ$ be a discrete finitely generated group. Let $\hat{M}\to T$ be a $Γ$-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary $Z$. We assume that $Γ\to \hat{M}\to \hat{M}/Γ$ is a Galois covering of a compact manifold with boundary. Let $(D^+ (θ))_{θ\in T}$ be a $Γ$-equivariant family of Dirac-type operators. Under the assumption that the boundary family is $L^2$-invertible, we define an index class in the K-theory of the cross-product algebra, $K_0 (C^0 (T)\rtimes_r Γ)$. If, in addition, $Γ$ is of polynomial growth, we define higher indeces by …
As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid $\cG_{H} M$ encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of a refined …
We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.
We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them almost regular Poisson structures and determine them completely. They include regular Poisson and log symplectic manifolds, as well as several other Poisson structures whose symplectic foliation presents singular…
We define and study, under suitable assumptions, the fundamental class, the index class and the rho class of a spin Dirac operator on the regular part of a spin stratified pseudomanifold. More singular structures, such as singular foliations, are also treated. We employ groupoid techniques in a crucial way; however, an effort has been made in order to make this article accessible to readers with only a minimal knowledge of groupoids. Finally, whenever appropriate, a comparison between classical microlocal methods and groupoids methods has been provided.
We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we der…
We develop here a concept of deformed algebras and their related groups through two examples. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \cite{Ma2013} is only an example among others, and how they often give rise to regular Frölicher Lie groups, and sometimes to Fréchet Lie groups, that are regular. The first example, indexed by $\mathbb{N},$ is obtained by deformations of the group of bounded classical pseudo-differential operators $Cl^{0,*}$ by algebras of (maybe unbo…
We prove the Fredholmness of Dirac operators of monopoles with Dirac-type singularities on oriented complete Riemannian $3$-folds, and we also calculate the $L^2$-indices of them.
We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained.
Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy $b$-groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. We further use the blup groupoids as the one above, and in particular its functoriality pr…
Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our construction is well defined under the weaker assumption of positive Novikov-Shubin invariant. We prove that this torsion form appears in a transgression formula, from which a non-commutative Riamannian-Roch-Grothendieck index formula follows.
In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with boundary.
The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and involves analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C*-algebra of symbols. There is a canonic…
We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach.
Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered b…
We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on its link. We explicitly calculate the stability index for cones on group orbits. We also describe the stability index for cones fibered by 2-planes over algebraic curves using the degree and genus of the curve and the spectrum of the Laplacian on the link. Finally we apply our r…
The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. Enlightening from Alain Connes' tangent groupoid proof of the index theorem and van Erp's research for the Heisenberg index theory on contact manifolds, we give an explicit construction of a series of maps, whose induced map in K-theory is the Heisenberg Atiyah-Singer index map on contact manifolds. Our methods derive from Higson's construction for symbol class in K-theory.
We show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, are either empty or smooth hyperkähler manifolds. Furthermore, we prove an index theorem and therefore compute the dimension of the moduli spaces.
We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are g…
Surfaces that evolve by mean curvature flow develop singularities. These singularities can be modeled by self-shrinkers, surfaces that shrink by dilations under the flow. Singularities modeled on classical self-shrinkers, namely spheres and cylinders, are stable under perturbations of the flow. In contrast, singularities modeled on other self-shrinkers, such as the Angenent torus, are unstable: perturbing the flow will generally change the kind of singularity. One can measure the degree of instability by computing the Morse index of the self-shrinker, viewed as a critical point of an appropria…
This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more general, stratified spaces. This book provides a self--contained treatment of the analysis of the heat equation and index theory for these operators. Major topics are short--time asymptotics, $η$-- and $ζ$--functions, (relative) index theorems. Another chapter is devoted to the discussion of deficiency indices and Dirac Schrödinger operators.
This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how this theorem is a special case of a much more general index theorem for subelliptic operators on contact manifolds. Finally we discuss the noncommutative topology that is employed in the proof o…
We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the Gromov-Lawson Conjecture for such manifolds in the case of particular type of singularities. We give an affirmative answer when such manifolds with singularities accept a metric of positive scalar curvature in terms of the index of the Dirac operator valued in the corresponding "K-theories with singularities". The key ideas are based on the construction due to Stolz, …
Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on ${\mathbb C}^n$ by a quasi-homogeneous polynomial $f$. Under some mild assumption on $f$, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time $t$. Then we prove a local index formula which expresses the Milnor number of $f$ by a Gaussian type integral. Furthermore, the heat kernel expansion provides spectral invariants of $f$. Especially, we define torsion type invaria…
Our goal in this work is to develop aspects of Bialynicki-Birula and Morse-Bott theory that can be extended from the classical setting of smooth manifolds to that of complex analytic spaces with a holomorphic $\mathbb{C}^*$ action. We extend prior results on existence of Bialynicki-Birula decompositions for compact, complex Kähler manifolds to non-compact complex manifolds and develop functorial properties of the Bialynicki-Birula decomposition, in particular with respect to blowup along a $\mathbb{C}^*$-invariant, embedded complex submanifold. We deduce the existence of a Bialynicki-Birula de…
