Positive scalar curvature, higher rho invariants and localization algebras

Xie, Zhizhang; Yu, Guoliang

doi:10.1016/j.aim.2014.06.001

Cited by 70 publications

(63 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To compare our construction with previous approaches, in the Appendix we describe our secondary invariants explicitly as ordinary complex K-theory classes in terms of projections and unitaries. As a consequence, we observe that our construction of the local index classes and ρ-invariants is essentially equivalent to Xie-Yu's [28].…”

Section: Introductionmentioning

confidence: 80%

“…Bordism classes of spin manifolds with positive scalar curvature metric as above form a group Pos spin * (BΓ) which is part of Stolz' positive scalar curvature sequence. In fact, in [16,28] a map from the Stolz sequence to the Higson-Roe sequence (6.1) is constructed and the delocalized APS-index theorem Corollary 6.6 is a central ingredient for that.…”

Section: Stability Of Higher Secondary Invariants On Closed Manifoldsmentioning

confidence: 99%

“…We describe the partial secondary local index classes as elements in K 0 (C * L,Z (X)) and K 1 (C * L,Z (X)). This is done in essentially the same way as Xie-Yu [28] define the local index class and the ρ-invariant. We show that this agrees with the elements defined in Subsection 4.2 up to a sign and a natural isomorphism K 0 (A ⊗Cl n ) ∼ = K n (A) for ungraded C * -algebras A.…”

Section: Appendix Explicit Descriptions In Terms Of Projections and mentioning

confidence: 99%

“…These secondary invariants have been the focus of intensive study in the recent past; see, for instance, [12,16,20,24,[26][27][28]. However, in standard applications, one takes X := M to be the universal covering of a closed spin manifold M and Γ = π 1 (M ).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Positive scalar curvature and product formulas for secondary index invariants

Zeidler¹

2016

J Topology

View full text Add to dashboard Cite

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature (upsc) outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the higher ρ-invariant of a metric with upsc as well as for the higher relative index of two metrics with upsc.Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized Atiyah-Patodi-Singer (APS)-index theorem of Piazza-Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the higher relative index. We also show that secondary invariants are stable with respect to direct products with aspherical manifolds that have fundamental groups of finite asymptotic dimension. Moreover, we construct examples of complete metrics with upsc on non-compact spin manifolds that can be distinguished up to concordance relative to subsets which are coarsely negligible in a certain sense.A technical novelty in this paper is that we use Yu's localization algebras in combination with the description of K-theory for graded C * -algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas.

show abstract

Section: Introductionmentioning

confidence: 80%

Section: Stability Of Higher Secondary Invariants On Closed Manifoldsmentioning

confidence: 99%

Section: Appendix Explicit Descriptions In Terms Of Projections and mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Positive scalar curvature and product formulas for secondary index invariants

Zeidler¹

2016

J Topology

View full text Add to dashboard Cite

show abstract

“…Remark 1.32. After the first publication of the present paper in the arXiv, Xie and Yu, in the preprint [45], treated the problem with a different method. They use Yu's localization algebras and an exterior product structure between K-homology and the analytic structure group to reduce the proof of the main result of this paper to the known behavior of the Khomology fundamental class under the Mayer-Vietoris boundary map.…”

Section: Mapping the Positive Scalar Curvature Sequence To Analysismentioning

confidence: 99%

Rho-classes, index theory and Stolz’ positive scalar curvature sequence

Piazza

Schick

2014

Journal of Topology

134

View full text Add to dashboard Cite

show abstract

Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

Weinberger

Xie

2020

Comm Pure Appl Math

Self Cite

View full text Add to dashboard Cite

Let X be a closed oriented connected topological manifold of dimension n ! 5. The structure group S TOP .X/ is the abelian group of equivalence classes of all pairs .f; M / such that M is a closed oriented manifold and f M 3 X is an orientation-preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant map defines a group homomorphism from the topological structure group S TOP .X/ of X to the analytic structure group K n .C £ L;0. z X/ / of X. Here z X is the universal cover of X, h 1 X is the fundamental group of X, and C £ L;0. z X/ is a certain C £-algebra. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected topological manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected homology manifold. As an application, we use the additivity of the higher rho invariant map to study nonrigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the algebraically reduced structure group of X by the number of torsion elements in 1 X. Here the algebraically reduced structure group of X is the quotient of S TOP .X/ modulo a certain action of self-homotopy equivalences of X. We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds.

show abstract

Positive scalar curvature, higher rho invariants and localization algebras

Cited by 70 publications

References 32 publications

Positive scalar curvature and product formulas for secondary index invariants

Positive scalar curvature and product formulas for secondary index invariants

Rho-classes, index theory and Stolz’ positive scalar curvature sequence

Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

Contact Info

Product

Resources

About