Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants

Deeley, Robin J.; Goffeng, Magnus

doi:10.1007/s00208-016-1365-6

Cited by 9 publications

(33 citation statements)

References 47 publications

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“…Building on the results of the current paper, we construct Chern characters in the relative case in . The construction in is similar to the one in [, Section 4]. The assembly maps we consider in this paper are for free actions and coincides with Baum–Connes' assembly mapping for proper actions when the involved groups are torsion‐free.…”

Section: Introductionmentioning

confidence: 93%

“…Proof The second statement of the lemma follows from the first: the choice of Dirac operators is from a path‐connected space and by making a choice of a continuous path of trivializing operators, the class

{normalΦ}_{cone} false(W, (E_{B_{2}}, F_{B_{1}}, α) false)

is well defined due to homotopy invariance of

K

‐theory. Such an argument is standard, see for instance [, Lemma 3.4; , Section 2.5]. To prove the first statement, we use the following elementary fact in

K

‐theory: suppose that

u

and

\overset{u}{true}

are unitaries in the unitalization of

C_{0} false((0, 1], B_{2} \otimes double-struckK false)

such that

u false(1 false) = \tilde{u} false(1 false)

.…”

Section: The Isomorphism Between the Geometric And Analytic Realizatimentioning

confidence: 99%

“…Additivity of the index and of the Cayley transform shows that

\begin{matrix} true \\ left Φ_{cone} (W, false(E_{B_{2}}, F_{B_{1}}, α false)) \\ left = Φ_{cone} (W, false(E_{B_{2}}, F_{B_{1}}, α false)) + Φ_{cone} (W_{0}, false({\tilde{E}}_{B_{2}} |_{W_{0}}, {\tilde{F}}_{B_{1}}, u false)) \\ left = Φ_{cone} (\partial Z, false({\tilde{E}}_{B_{2}}, \emptyset, \emptyset false)) = 0, \end{matrix}

where the last identity follows from the bordism invariance of the index. To show that

Φ_{cone}

respects vector bundle modification, we use of a trick from (also see ). Let

(W, false(E_{B_{2}}, F_{B_{1}}, α false), false(D_{E}, D_{F} false), A)

be a cycle with Dirac operators and trivializing operator and

V \to W

a spin

^{c}

‐vector bundle of even rank.…”

Section: The Isomorphism Between the Geometric And Analytic Realizatimentioning

confidence: 99%

See 2 more Smart Citations

Relative geometric assembly and mapping cones, part I: the geometric model and applications

Deeley

Goffeng

2018

Journal of Topology

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Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric K-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (for example, the localized index and higher Atiyah-Patodi-Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to the work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map implies homotopy invariance of the relative higher signatures of a manifold with boundary.

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Section: Introductionmentioning

confidence: 93%

{normalΦ}_{cone} false(W, (E_{B_{2}}, F_{B_{1}}, α) false)

is well defined due to homotopy invariance of

K

‐theory. Such an argument is standard, see for instance [, Lemma 3.4; , Section 2.5]. To prove the first statement, we use the following elementary fact in

K

‐theory: suppose that

u

and

\overset{u}{true}

are unitaries in the unitalization of

C_{0} false((0, 1], B_{2} \otimes double-struckK false)

such that

u false(1 false) = \tilde{u} false(1 false)

.…”

Section: The Isomorphism Between the Geometric And Analytic Realizatimentioning

confidence: 99%

“…Additivity of the index and of the Cayley transform shows that

\begin{matrix} true \\ left Φ_{cone} (W, false(E_{B_{2}}, F_{B_{1}}, α false)) \\ left = Φ_{cone} (W, false(E_{B_{2}}, F_{B_{1}}, α false)) + Φ_{cone} (W_{0}, false({\tilde{E}}_{B_{2}} |_{W_{0}}, {\tilde{F}}_{B_{1}}, u false)) \\ left = Φ_{cone} (\partial Z, false({\tilde{E}}_{B_{2}}, \emptyset, \emptyset false)) = 0, \end{matrix}

where the last identity follows from the bordism invariance of the index. To show that

Φ_{cone}

respects vector bundle modification, we use of a trick from (also see ). Let

(W, false(E_{B_{2}}, F_{B_{1}}, α false), false(D_{E}, D_{F} false), A)

be a cycle with Dirac operators and trivializing operator and

V \to W

a spin

^{c}

‐vector bundle of even rank.…”

Section: The Isomorphism Between the Geometric And Analytic Realizatimentioning

confidence: 99%

See 1 more Smart Citation

Relative geometric assembly and mapping cones, part I: the geometric model and applications

Deeley

Goffeng

2018

Journal of Topology

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“…For now, we allow W to be noncompact; we will mainly focus on the case of W is complete or a compact manifold with boundary. Further details on this setup can be found in [12,16,35]. We tacitly assume that all structures (the metric on W , E B and its connection ∇ E ) are of product type near ∂W .…”

Section: Dirac Operators On Manifoldsmentioning

confidence: 99%

The bordism group of unbounded KK-cycles

2018

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Abstract. We consider Hilsum's notion of bordism as an equivalence relation on unbounded KK-cycles and study the equivalence classes. Upon fixing two C * -algebras, and a * -subalgebra dense in the first C * -algebra, a Z/2Z-graded abelian group is obtained; it maps to the Kasparov KK-group of the two C * -algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first C * -algebra is the complex numbers (i.e., for K-theory) and is a split surjection if the first C * -algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense * -subalgebra.

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“…The motivations for such an investigation come from at least three sources. Firstly, in recent years, the importance of secondary invariants in geometry has been realised (see for instance [30,39,41,46,47]), and the need to consider Dirac-type operators on manifolds with boundary in their study [18]. Secondly, having a relative spectral triple representing a K-homology class x should facilitate the behaviour of n. This is done in the assumptions in Definition 3.1 on page 14, and these must be checked in examples.…”

Section: Introductionmentioning

confidence: 99%

Boundaries, spectral triples and $K$-homology

Forsyth

Goffeng

Mesland

et al. 2019

J. Noncommut. Geom.

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This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal J ⊳ A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, θ-deformations and Cuntz-Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in K-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the K-homological boundary. Thus we abstract the proof of Baum-Douglas-Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.computation of the image ∂x of x under the K-homology boundary map ∂: a difficult problem in general, [6]. Finally, internal to non-commutative geometry there is the more philosophical question of what should be the non-commutative analogue of a manifold with boundary.Historically, attempts to incorporate manifolds with boundary into non-commutative geometry have either imposed self-adjoint boundary conditions on a Dirac operator (like APS), which generically collapse the boundary to a point, for example [5,31], or they have made the Dirac operator selfadjoint by pushing the boundary to infinity. In fact APS boundary conditions are closely related to the latter framework. More general constructions are possible, for instance [33].The main idea in this paper is to loosen the self-adjointness condition appearing in a spectral triple, requiring only a symmetric operator with additional analytic properties relative to an ideal. Sections 2 and 3 and the appendices are based on results from the Ph.D thesis of the first named author, [20]. The main resultsOur approach is inspired by the work of Baum-Douglas-Taylor [6], and the main players in this paper are relative spectral triples and Kasparov modules, introduced in Section 2. We work with a Z/2graded C * -algebra A and a closed graded * -ideal J ⊳ A. Loosely speaking, a relative spectral triple for J A is a triple (J A, H, D) satisfying the usual axioms of a spectral triple for the dense * -subalgebra A ⊆ A save the fact that the odd operator D is symmetric and j Dom(For the precise definition, see Definition 2.1 on page 5, and we follow the definition with numerous examples. The first important result is that relative spectral triples give relative Fredholm modules, and so relative K-homology classes. Theorem (Bounded transforms of relative spectral triples). Let...

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Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants

Cited by 9 publications

References 47 publications

Relative geometric assembly and mapping cones, part I: the geometric model and applications

Relative geometric assembly and mapping cones, part I: the geometric model and applications

The bordism group of unbounded KK-cycles

Boundaries, spectral triples and $K$-homology

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