In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdef... more In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdefined for a conical pseudomanifold and the Poincar\'e duality in $K$-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate and natural presentation of the notion of symbols on a manifold generalizes right away
This paper is devoted to the study of Poincar\'e duality in K-theory for general stratified p... more This paper is devoted to the study of Poincar\'e duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\fS$ of a topological space $X$ and we define a groupoid $T^{\fS}X$, called the $\fS$-tangent space. This groupoid is made of different pieces encoding the tangent spaces of the strata, and these pieces are glued
Geometric and Topological Methods for Quantum Field Theory, 2010
These lecture notes are mainly devoted to a proof using groupoids and KK-theory of Atiyah-Singer ... more These lecture notes are mainly devoted to a proof using groupoids and KK-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, C * -algebras, KKtheory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudomanifolds.
Proceedings of the London Mathematical Society, 2006
We describe all multiplicative determinants on the pathwise connected component of identity in th... more We describe all multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold, that are continuous along continuous paths and the restriction to zero order operators of which is of class C 1 . This boils down to a description of all traces on zero order classical pseudodifferential operators, which turn out to be linear combinations of the Wodzicki residue [W] and leading symbol traces introduced in [PR1], both of which are continuous. Consequently, multiplicative determinants are parametrized by the residue determinant [W87, Sc] and a new "leading symbol determinant", both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator.
Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2009
, a notion of noncommutative tangent space is associated with a conical pseudomanifold and the Po... more , a notion of noncommutative tangent space is associated with a conical pseudomanifold and the Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate presentation of the notion of symbols on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret the Poincaré duality in the singular setting as a noncommutative symbol map.
We associate to a pseudomanifold X with a conical singularity a differentiable groupoid G which p... more We associate to a pseudomanifold X with a conical singularity a differentiable groupoid G which plays the role of the tangent space of X : We construct a Dirac element and a dual Dirac element which induce a K-duality between the C Ã -algebras C Ã ðGÞ and CðX Þ: This is a first step toward an index theory for pseudomanifolds. r 2004 Elsevier Inc. All rights reserved.
This paper continues the project started in where Poincaré duality in K -theory was studied for s... more This paper continues the project started in where Poincaré duality in K -theory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification S of a topological space X and we define a groupoid T S X , called the S-tangent space. This groupoid is made of different pieces encoding the tangent spaces of strata, and these pieces are glued into the smooth noncommutative groupoid T S X using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that C * (T S X) is Poincaré dual to C(X), in other words, the S-tangent space plays the role in K -theory of a tangent space for X . 58B34, 46L80, 19K35, 58H05, 57N80; 19K33, 19K56, 58A35
We associate to a pseudomanifold X with an isolated singularity a differentiable groupoid G which... more We associate to a pseudomanifold X with an isolated singularity a differentiable groupoid G which plays the role of the tangent space of X. We construct a Dirac element D and a Dual Dirac element λ which induce a Poincaré duality in K-theory between the C * -algebras C(X) and C * (G). c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1998
Sur une pseudovariCtC de dimension paire & une singularit6 conique isolke, des triplets spectraux... more Sur une pseudovariCtC de dimension paire & une singularit6 conique isolke, des triplets spectraux sont construits B partir d'une classe d'opkrateurs diffkrentiels elliptiques de type Fuchs, contenant les opkrateurs de Dirac ?I coefficients dans des fibr& plats dans la direction radiale. Ces derniers engendrent, sous une hypothkse raisonnable, le groupe de K-homologie pair tensoris par Q: de la pseudovari6tC et leur caract& de Chern est calculk. 0 AcadCmie des Sciences/Elsevier. Paris Spectral triples for pseudomanifolds with isolated singularity Abstract. We use elliptic operators of Fuchs ~ypr on an el,erl-dimen.sional pseudoman~fold with an isoluted singularity to c'onstruc~t spectral tripbs. This ~.Iu.ss of operators inc1llde.s Dirac operators with coeficknts in flrrt bundles in the radial direction and, under Some hypothesis, these operators generate the even K-homolog,y group tensori:ed by C, of the pseudoman~fbld. Moreover, their Chern rharacter is computed.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
We define an analytical index map and a topological index map for conical pseudomanifolds. These ... more We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M . A main new ingredient in our proof is a non-commutative algebra that plays in our setting the role of C 0 (T * M ). We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in K-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.
In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdef... more In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdefined for a conical pseudomanifold and the Poincar\'e duality in $K$-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate and natural presentation of the notion of symbols on a manifold generalizes right away
This paper is devoted to the study of Poincar\'e duality in K-theory for general stratified p... more This paper is devoted to the study of Poincar\'e duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\fS$ of a topological space $X$ and we define a groupoid $T^{\fS}X$, called the $\fS$-tangent space. This groupoid is made of different pieces encoding the tangent spaces of the strata, and these pieces are glued
Geometric and Topological Methods for Quantum Field Theory, 2010
These lecture notes are mainly devoted to a proof using groupoids and KK-theory of Atiyah-Singer ... more These lecture notes are mainly devoted to a proof using groupoids and KK-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, C * -algebras, KKtheory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudomanifolds.
Proceedings of the London Mathematical Society, 2006
We describe all multiplicative determinants on the pathwise connected component of identity in th... more We describe all multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold, that are continuous along continuous paths and the restriction to zero order operators of which is of class C 1 . This boils down to a description of all traces on zero order classical pseudodifferential operators, which turn out to be linear combinations of the Wodzicki residue [W] and leading symbol traces introduced in [PR1], both of which are continuous. Consequently, multiplicative determinants are parametrized by the residue determinant [W87, Sc] and a new "leading symbol determinant", both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator.
Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2009
, a notion of noncommutative tangent space is associated with a conical pseudomanifold and the Po... more , a notion of noncommutative tangent space is associated with a conical pseudomanifold and the Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate presentation of the notion of symbols on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret the Poincaré duality in the singular setting as a noncommutative symbol map.
We associate to a pseudomanifold X with a conical singularity a differentiable groupoid G which p... more We associate to a pseudomanifold X with a conical singularity a differentiable groupoid G which plays the role of the tangent space of X : We construct a Dirac element and a dual Dirac element which induce a K-duality between the C Ã -algebras C Ã ðGÞ and CðX Þ: This is a first step toward an index theory for pseudomanifolds. r 2004 Elsevier Inc. All rights reserved.
This paper continues the project started in where Poincaré duality in K -theory was studied for s... more This paper continues the project started in where Poincaré duality in K -theory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification S of a topological space X and we define a groupoid T S X , called the S-tangent space. This groupoid is made of different pieces encoding the tangent spaces of strata, and these pieces are glued into the smooth noncommutative groupoid T S X using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that C * (T S X) is Poincaré dual to C(X), in other words, the S-tangent space plays the role in K -theory of a tangent space for X . 58B34, 46L80, 19K35, 58H05, 57N80; 19K33, 19K56, 58A35
We associate to a pseudomanifold X with an isolated singularity a differentiable groupoid G which... more We associate to a pseudomanifold X with an isolated singularity a differentiable groupoid G which plays the role of the tangent space of X. We construct a Dirac element D and a Dual Dirac element λ which induce a Poincaré duality in K-theory between the C * -algebras C(X) and C * (G). c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1998
Sur une pseudovariCtC de dimension paire & une singularit6 conique isolke, des triplets spectraux... more Sur une pseudovariCtC de dimension paire & une singularit6 conique isolke, des triplets spectraux sont construits B partir d'une classe d'opkrateurs diffkrentiels elliptiques de type Fuchs, contenant les opkrateurs de Dirac ?I coefficients dans des fibr& plats dans la direction radiale. Ces derniers engendrent, sous une hypothkse raisonnable, le groupe de K-homologie pair tensoris par Q: de la pseudovari6tC et leur caract& de Chern est calculk. 0 AcadCmie des Sciences/Elsevier. Paris Spectral triples for pseudomanifolds with isolated singularity Abstract. We use elliptic operators of Fuchs ~ypr on an el,erl-dimen.sional pseudoman~fold with an isoluted singularity to c'onstruc~t spectral tripbs. This ~.Iu.ss of operators inc1llde.s Dirac operators with coeficknts in flrrt bundles in the radial direction and, under Some hypothesis, these operators generate the even K-homolog,y group tensori:ed by C, of the pseudoman~fbld. Moreover, their Chern rharacter is computed.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
We define an analytical index map and a topological index map for conical pseudomanifolds. These ... more We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M . A main new ingredient in our proof is a non-commutative algebra that plays in our setting the role of C 0 (T * M ). We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in K-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.
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Papers by Jean-marie Lescure