Surgery Theory
Winter Term 2015/2016
lecture by Prof. Dr. H. Reich, Prof. Dr E. Vogt, Dr. D. Egas Santander, Dr. F. Levikov
-
Time and place: Wednesdays 12 - 14 in SR210, Arminallee 3.
(NEW: From the second week onwards the lecture might be shifted to
Tuesdays 14 -16. If you are interested, please make sure you attend the first meeting or email Filipp Levikov!) - Exercise Session: tba
Course Overview
Surgery theory is a very successful method for classifying high-dimensional manifolds. For our purposes the manifolds will be differentiable and closed while high-dimensional will mean at least 5-dimensional. Surgery theory dates back to Milnor's discovery of exotic spheres - closed manifolds which are homemorphic but not diffeomorphic to the standard sphere - the subsequent classification of exotic spheres by Kervaire and Milnor (see the companion seminar) and the seminal work of Browder, Novikov, Sullivan and Wall.
The general idea is to tell two manifolds M and N apart which are assumed to be homotopy equivalent. By cutting out and reattaching handles, the manifolds M and N can be made h-cobordant once certain algebraic obstructions vanish. If the manifolds are simply connected the h-cobordism theorem by Smale implies that they are already diffeomorphic. In the general case, the fundamental group gives rise to an algebraic obstruction - the Whitehead torsion - whose vanishing implies that M and N are diffeomorphic. This is an instance of the s-cobordism theorem by Barden-Mazur-Stallings which will be the starting point of the lecture course.
Vorträge/Talks (in the semester break)
We will continue with talks during the semester break to cover parts of section 4 of Crowley-Lück-Macko. We will meet on the given dates at 14:15-15:45 in room A7SRE31.
| Termin | Titel | Vortragende(r) |
|---|---|---|
| 14.10. | Introduction | Elmar |
| 20.10. | s-Cobordism Theorem | Filipp |
| 27.10. | s-Cobordism Theorem | Filipp |
| 03.11. | s-Cobordism Theorem | Filipp |
| 10.11. | Whitehead Torsion | Filipp |
| 17.11. | Whitehead Torsion | Filipp |
| 24.11. |
Poincaré Duality and Geometric Poincaré Complexes | Daniela |
| 01.12. |
Poincaré Duality and Geometric Poincaré Complexes | Daniela |
| 08.12. | Spherical Fibrations, Spivak Normal Fibration | Holger |
| 15.12. | Spherical Fibrations, Spivak Normal Fibration | Holger |
| 05.01. | Spivak Normal Fibration | Elmar |
| 12.01. | The Surgery Step | Daniela |
| 19.01. | The Surgery Step | Filipp |
| 26.01. | Normal Maps, Normal Structures | Elmar |
| 23.02. | Intersections of Immersions (Section 4.2) | Elmar/Daniela |
| 25.02. | Surgery kernels in the closed case (4.3.1) | Peter |
| 29.02. | Symmetric Forms and Surgery Kernels (4.3.3) | Filipp |
| 03.03. | Quadratic Gorms and Surgery Kernels (4.3.4) | Daniel |
| 14.03. | Even dimensional L-groups (4.4) | Mark |
| 15.03. | The Surgery Obstruction in Even Dimensions (4.5) | Holger |
References
Diarmuid Crowley, Wolfgang Lück und Tibor Macko Surgery Theory: Foundation, preprint of a book project, available online.
Andrew Ranicki, Algebraic and Geometric Surgery, OUP 2002, also available from the author's website.
C. T. C. Wall, Surgery on compact manifolds, AMS MSM 69, 1999, 2nd ed. edited by A. Ranicki, also available from the editor's website.