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 highdimensional manifolds. For our purposes the manifolds will be differentiable and closed while highdimensional will mean at least 5dimensional. 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 hcobordant once certain algebraic obstructions vanish. If the manifolds are simply connected the hcobordism 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 scobordism theorem by BardenMazurStallings 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 CrowleyLückMacko. We will meet on the given dates at 14:1515:45 in room A7SRE31.
Termin  Titel  Vortragende(r) 

14.10.  Introduction  Elmar 
20.10.  sCobordism Theorem  Filipp 
27.10.  sCobordism Theorem  Filipp 
03.11.  sCobordism 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 Lgroups (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.