Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model

Deeley, Robin J.; Goffeng, Magnus

doi:10.48550/arxiv.1308.5990

Cited by 4 publications

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“…Once we have at our disposal Proposition 2.27, together with its proof, it is elementary to check that the reduction to the cylinder proceeds exactly as in [16]. Thus (1) We establish that P 0 χ(D + C)…”

Section: Reduction To the Cylindermentioning

confidence: 83%

“…Similarly, the analytic structure group S 0 (Γ) is, by definition, the group K 1 (D * Γ ). In [1], Deeley and Goffeng have introduced a geometric structure group S geo 1 (Γ), with cycles defined à la Baum-Douglas. Using the results of the present paper, Deeley and Goffeng have proved that their geometric definition is isomorphic to the analytic one given by Higson and Roe.…”

Section: Introductionmentioning

confidence: 99%

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The surgery exact sequence, K-theory and the signature operator

Piazza

Schick

2016

AKT

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The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.Comment: 29 pages, AMS-LaTeX; v2: small corrections and (hopefully) improved exposition, as suggested by the referee. Final version, to appear in Annals of K-Theor

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Section: Reduction To the Cylindermentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

The surgery exact sequence, K-theory and the signature operator

Piazza

Schick

2016

AKT

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“…Firstly, it is the analytic analog of the surgery exact sequence and secondly, there is a precise relationship between the two exact sequences (see [20,21,22,41] for details); we also discuss this relationship in Section 5. We will return to the analytic surgery exact sequence in the context of the geometric model from [11] in Section 2.…”

Section: Preliminariesmentioning

confidence: 99%

“…This paper is the last in a series of three whose topic is the construction of a geometric (i.e., Baum-Douglas type) realization of the analytic surgery group of Higson and Roe [20,21,22]. The first paper in the series [11] dealt with the geometric cycles, the associated six term exact sequence, and the geometrically defined secondary invariants. In the second paper [12], the relationship between these geometrically defined secondary invariants and analytically defined secondary invariants was developed−mainly focusing on the relative η-invariant.…”

Section: Introductionmentioning

confidence: 99%

“…Here K * (D * ( X) Γ ) is Higson-Roe's analytic surgery group [20,21,22], explained further in Subsection 1.1, and j X the natural mapping K * (C * r (Γ)) → K * (D * ( X) Γ ). The triple (W, ξ, f ) is a geometric surgery cycle for the Mishchenko bundle L X → X from [11] and (W, Ξ, f ) denotes the cycle (W, ξ, f ) with further analytic data fixed. A short review of geometric cycles for surgery and the construction of λ can be Date: September 21, 2018. found in Subsection 2.1.…”

Section: Introductionmentioning

confidence: 99%

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

Deeley

Goffeng

2016

Math. Ann.

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We apply the geometric analog of the analytic surgery group of Higson and Roe to the relative η-invariant. In particular, by solving a Baum-Douglas type index problem, we give a "geometric" proof of a result of Keswani regarding the homotopy invariance of relative η-invariants. The starting point for this work is our previous constructions in "Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model".Date: September 24, 2018.

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Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model

Cited by 4 publications

References 0 publications

The surgery exact sequence, K-theory and the signature operator

The surgery exact sequence, K-theory and the signature operator

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

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

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