Among the basic invariants of closed oriented manifolds are signature invariants, taking values in suitable L-groups. The surgery exact sequence computes the structure set (which gives information about all manifolds of a given homotopy type) in particular in terms of such L-groups. An important open question is the Novikov conjecture, asserting that certain higher signatures are homotopy invariants.
On the other hand, signature operators have higher indices in the K-theory of C*-algebras, and have particularly nice properties (oriented homotopy invariance). Via the Baum-Connes conjecture, this can be used to prove the Novikov conjecture.
However, Higson and Roe propose a much more direct comparison between the two fields; the surgery exact sequence should have a parallel exact sequence in K-theory of C*-algebras, with a comparison map between those two.
The goal of the seminar is to understand this circle of ideas, following in particular the corresponding three papers of Higson and Roe.
|1||Review of Surgery theory 1||(HR3), 4, (L), (R), (W)||TS||18.04.|
|2||Review of Surgery theory 2||(L), (R), (W)||TS||25.04.|
|3||Review of Surgery theory 3||(L), (R), (W), (KR)||TS||02.05.|
|4||Analysis on Hilbert modules||(La)||A||09.05.|
|5||Signatures of Poincare Hilbert complexes||(HR1) 1-4||A||16.05.|
|6||Analytically controlled complexes||(HR1) 5||A||23.05.|
|7||Abstract bordism invariance||(HR1) 6-7||A||30.05.|
|8||Geometric Hilbert-Poincare complexes||(HR2) 2-3||GA||06.06.|
|9||Smooth manifolds as examples 1||(HR2) 4-5||GA||13.06.|
|10||Smooth manifolds as examples 2||(HR2) 4-5||GA||20.06.|
|11||Paschke Duality and K-homology||SA||27.06.|
|12||The analytic surgery sequence||(HR3) 1-2||SA||04.07.|
|13||Mapping surgery to analysis||(HR3) 3,5-6||SA||11.07.|
|S||Surgery||Ulrich Pennig, Nora, Andreas Lochmann||Uli, Thomas|
|A||Analysis||Ingo, Behnam, Ansgar, Moritz||Ralf|
|GA||Geometry||Andreas Thom, Roman|
|SA||final goal||Ralf, Thomas, Ulrich Bunke|
It might be useful to take more time for the paper, in particular to add discussions of the (analytic) background.
The surgery talks should also come to the difference/parallelism between the surgery theory for differentiable manifolds and the one for topological manifolds.
We now plan to prepare for the surgery talks with some kind of a surgery lecture course, open for other interested participants. This course will be open for other interested participants and will meet in the last weeks before the start of the sememster, and will be supervised jointly (or alternatingly) by Ulrich Bunke and Thomas Schick.
(HR1) Higson and Roe: Mapping surgery to analysis I
(HR2) Higson and Roe: Mapping surgery to analysis II
(HR3) Higson and Roe: Mapping surgery to analysis III
(HS) Hilsum and Skandalis; Invariance par homotopie de la signature a coefficients dans un fibre presque plat
(KL) Kreck and Lück; The Novikov conjecture: Geometry and Algebra, Birkhäuser
(L) Lück; A basic introduction to surgery theory
(La) Lance: Hilbert C*-modules, LMS
(PS) Piazza, Schick; Bordism, Rho-invariants and the Baum-Connes conjecture
(R) Ranicki; Algebraic and geometric surgery, Cambridge University Press
(W) Wall; Surgery on compact manifolds