| 1. |
- Deeley, R.J., et al.
(författare)
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Applying geometric K-cycles to fractional indices
- 2017
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Ingår i: Mathematische Nachrichten. - : Wiley. - 1522-2616 .- 0025-584X. ; 290:14-15, s. 2207-2233
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Tidskriftsartikel (refereegranskat)abstract
- A geometric model for twisted K-homology is introduced. It is modeled after the Mathai–Melrose–Singer fractional analytic index theorem in the same way as the Baum–Douglas model of K-homology was modeled after the Atiyah–Singer index theorem. A natural transformation from twisted geometric K-homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted K-homology in this model is an isomorphism for torsion twists on a finite CW-complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study T-duality for geometric cycles.
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| 2. |
- Deeley, R.J., et al.
(författare)
-
Realizing the analytic surgery group of Higson and Roe geometrically, part I: the geometric model
- 2017
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Ingår i: Journal of Homotopy and Related Structures. - : Springer Science and Business Media LLC. - 2193-8407 .- 1512-2891. ; 12:1, s. 109-142
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Tidskriftsartikel (refereegranskat)abstract
- We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of η-type invariants from the geometric group (with respect to assembly for the full group C ∗ -algebra) to the real numbers.
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| 3. |
- Deeley, R.J., et al.
(författare)
-
Realizing the analytic surgery group of Higson and Roe geometrically part II: relative η -invariants
- 2016
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 366:3-4, s. 1319-1363
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Tidskriftsartikel (refereegranskat)abstract
- 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”.
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| 4. |
- Deeley, R.J., et al.
(författare)
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Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants
- 2016
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 366:3-4, s. 1513-1559
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Tidskriftsartikel (refereegranskat)abstract
- 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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| 5. |
- Deeley, R. J., et al.
(författare)
-
SMALE SPACE C*-ALGEBRAS HAVE NONZERO PROJECTIONS
- 2020
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 148:4, s. 1625-1639
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Tidskriftsartikel (refereegranskat)abstract
- The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.
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