Surgery theory is a very successful method for classifying high-dimensional manifolds. For our purposes the manifolds will be differentiable and closed while ... read more
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.