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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.

TerminTitelVortragende(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.