Realizing the analytic surgery group of Higson and Roe geometrically part II: relative $$\eta $$ η -invariants

Deeley, Robin J.; Goffeng, Magnus

doi:10.1007/s00208-016-1364-7

Cited by 6 publications

(8 citation statements)

References 38 publications

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“…The geometric model [17,18,19] allows for explicit geometric invariants and, in particular, Chern characters defined at the level of cycles that are related to the higher index theorems of Wahl [53] and Lott [40]. The results in the present paper for the assembly of relative K-groups are closely modelled on the results in [19] for Higson-Roe's surgery group.…”

Section: Introductionmentioning

confidence: 87%

“…Remark . For

C^{*}

‐coefficients

A

one uses

A

‐Clifford bundles as in [, Definition 2.9]. The reader is referred to [, Section 2.3] for the isomorphism of the oriented model with the spin

^{c}

‐model for geometric

K

‐homology with

C^{*}

‐algebra coefficients.…”

Section: The Relative Assembly Map As a Geometrically Defined Mapmentioning

confidence: 99%

“…In this section we prove the following theorem: We now turn to the proof of Theorem 5.1. The idea is to use the absolute case to prove the commutativity of the diagram (20). It remains to verify that the assumptions in Theorem 5.1 imply the commutativity of (20).…”

Section: Comparing Assembly Mapsmentioning

confidence: 99%

“…The notion of connection in the context of geometric cycles for surgery was defined in [19,Definition 4.10]. Connections in that context are more complicated due to the existence of "non-easy cycles" (Such cycles are needed in the surgery case because the Mishchenko bundle on the boundary need not extend, see [18,Introduction] for more on this). The distinction between connections and total connections was not emphasized in [19].…”

Section: Chern Characters On Relative K-theorymentioning

confidence: 99%

“…We need to show that y vanishes (rationally). Since the diagram (20) commutes in the absolute case, δ(y…”

mentioning

confidence: 99%

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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: 87%

“…Remark . For

C^{*}

‐coefficients

A

one uses

A

‐Clifford bundles as in [, Definition 2.9]. The reader is referred to [, Section 2.3] for the isomorphism of the oriented model with the spin

^{c}

‐model for geometric

K

‐homology with

C^{*}

‐algebra coefficients.…”

Section: The Relative Assembly Map As a Geometrically Defined Mapmentioning

confidence: 99%

Section: Comparing Assembly Mapsmentioning

confidence: 99%

Section: Chern Characters On Relative K-theorymentioning

confidence: 99%

“…We need to show that y vanishes (rationally). Since the diagram (20) commutes in the absolute case, δ(y…”

mentioning

confidence: 99%

See 3 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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Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants

Deeley

Goffeng

2016

Math. Ann.

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We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on "Realizing the analytic surgery group of Higson and Roe geometrically" with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a "delocalized Chern character", from which Lott's higher delocalized ρ-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature sequence to the geometric model of Higson-Roe's analytic surgery exact sequence.

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Adiabatic groupoid and secondary invariants in K-theory

Zenobi¹

2019

Advances in Mathematics

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Realizing the analytic surgery group of Higson and Roe geometrically part II: relative $$\eta $$ η -invariants

Cited by 6 publications

References 38 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

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

Adiabatic groupoid and secondary invariants in K-theory

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