Homotopy classification of elliptic operators on stratified manifolds

Назайкинский, В. Е.; Savin, Anton; Sternin, B. Yu.

doi:10.1134/s1064562406030252

Cited by 16 publications

(42 citation statements)

References 38 publications

(88 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Hence a standard construction (e.g., see [4]) defines the corresponding class in analytic K-homology of M, which we denote by…”

Section: Geometrymentioning

confidence: 99%

“…Consider the zero-order pseudodifferential operator D = ∆ −1/2 ∂M • (D ∂M ⊗ 1 E| ∂M ) on the total space ∂M of π as a pseudodifferential operator on X with operatorvalued symbol in the sense of [17]. Then by the generalized Luke formula [4] we obtain…”

Section: Lemma 42 One Hasmentioning

confidence: 99%

“…The corresponding obstruction was computed in [4] and is a generalization of the Atiyah-Bott obstruction [5] in the theory of classical boundary value problems. For the case in which the obstruction vanishes, we construct a Fredholm realization of the Dirac operator in the class of boundary value problems introduced in [6].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Poincaré isomorphism in K-theory on manifolds with edges

2010

Self Cite

View full text Add to dashboard Cite

The aim of this paper is to construct the Poincaré isomorphism in K-theory on manifolds with edges. We show that the Poincaré isomorphism can naturally be constructed in the framework of noncommutative geometry. More precisely, to a manifold with edges we assign a noncommutative algebra and construct an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges viewed as a compact topological space.

show abstract

“…Hence a standard construction (e.g., see [4]) defines the corresponding class in analytic K-homology of M, which we denote by…”

Section: Geometrymentioning

confidence: 99%

Section: Lemma 42 One Hasmentioning

confidence: 99%

See 1 more Smart Citation

On the Poincaré isomorphism in K-theory on manifolds with edges

2010

Self Cite

View full text Add to dashboard Cite

show abstract

“…4, where a large parameter is introduced into the Atiyah-Singer formula 1 while the index formula in terms of noncommutative geometry is obtained from the Atiyah-Singer formula as the parameter tends to infinity. Note that the ideas behind the technique used in this section are related to asymptotical homomorphisms widely used in noncommutative geometry and elliptic theory (see, e.g., [7,9,11]). …”

Section: Introductionmentioning

confidence: 99%

On the Index Formula for an Isometric Diffeomorphism

2014

Self Cite

View full text Add to dashboard Cite

We give an elementary solution to the problem of the index of elliptic operators associated with shift operator along the trajectories of an isometric diffeomorphism of a smooth closed manifold. This solution is based on index-preserving reduction of the operator under consideration to some elliptic pseudo-differential operator on a higher-dimension manifold and on the application of the Atiyah-Singer formula. The final formula of the index is given in terms of the symbol of the operator on the original manifold.

show abstract

“…But the specific issues about Poincaré duality, bivariant K -theory, topological index maps and [28,26,16,15,32,30].…”

Section: Introductionmentioning

confidence: 99%

K–duality for stratified pseudomanifolds

Debord

Lescure

2009

Geom. Topol.

View full text Add to dashboard Cite

This paper continues the project started in [13] 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 IntroductionThis paper takes place in a longstanding project aiming to study index theory and related questions on stratified pseudomanifolds using tools and concepts from noncommutative geometry.The key observation at the beginning of this project is that in its K -theoritic form, the Atiyah-Singer index theorem [2] involves ingredients that should survive to the singularities allowed in a stratified pseudomanifold. This is possible, from our opinion, as soon as one accepts reasonable generalizations and new presentation of certain classical objects on smooth manifolds, making sense on stratified pseudomanifolds.The first instance of these classical objects that need to be adapted to singularities is the notion of tangent space. Since index maps in [2] are defined on the K -theory of the tangent spaces of smooth manifolds, one must have a similar space adapted to stratified pseudomanifolds. Moreover, such a space should satisfy natural requirements. It should coincide with the usual notion on the regular part of the pseudomanifold and incorporate in some way copies of usual tangent spaces of strata, while keeping enough smoothness to allow interesting computations. Moreover, it should be Poincaré dual In [13], we introduced a candidate to be the tangent space of a pseudomanifold with isolated conical singularities. It appeared to be a smooth groupoid, leading to a noncommutative C * -algebra, and we proved that it fulfills the expected K -duality.In [22], the second author interpreted the duality proved in [13] as a principal symbol map, thus recovering the classical picture of Poincaré duality in K -theory for smooth manifolds. This interpretation used a notion of noncommutative elliptic symbols, which appeared to be the cycles of the K -theory of the noncommutative tangent space.In [14], the noncommutative tangent space together with other deformation groupoids was used to construct analytical and topological index maps, and their equality was proved. As expected, these index maps are straight generalizations of those of [2] for manifolds.The present paper is devoted to the construction of the noncommutative tangent space for a general stratified pseudomanifold and the proof of ...

show abstract

Homotopy classification of elliptic operators on stratified manifolds

Cited by 16 publications

References 38 publications

On the Poincaré isomorphism in K-theory on manifolds with edges

On the Poincaré isomorphism in K-theory on manifolds with edges

On the Index Formula for an Isometric Diffeomorphism

K–duality for stratified pseudomanifolds

Contact Info

Product

Resources

About