Der Kurs dient der Vorbereitung auf die schriftliche Prüfung in Analysis im 1. Staatsexamen (Lehramt Gymnasium). Anhand früherer Examensaufgaben sollen die erforderlichen Kenntnisse aus der Funktionentheorie, der reellen Analysis und aus der Theorie der (gewöhnlichen) Differentialgleichungen wiederholt und die wesentlichen Techniken zum Lösen der Aufgaben eingeübt werden.
Simple-homotopy theory asks whether any homotopy equivalence of finite CW-complexes can be expressed as a composition of some elementary geometric moves on the cells (called elementary expansions and elementary collapses). It turns out that the answer to this question is highly dependent on the fundamental group, and the non-simpleness of a homotopy equivalence can be detected by an element of the Whitehead group of the fundamental group, which is a certain K-theoretic invariant. This provides for example tools to distinguish lens spaces which are homotopy equivalent, but not homeomorphic.
In the later talks of the seminar, we will see how this theory comes to bear in understanding h-cobordisms between closed manifolds, ie cobordisms with the property that the inclusion of each boundary component is a homotopy equivalence. The s-cobordism theorem asserts that such an h-cobordism is isomorphic to a cylinder on one of the boundaries precisely if the inclusion of that boundary component is a simple homotopy equivalence.
This course covers some applications of algebraic K-theory (in
particular the class group K_0) in geometric/algebraic topology.
We will primarily cover the Wall finiteness obstruction which is a
K-theoretic invariant designed to detect whether certain topological
spaces are homotopy equivalent to finite CW-complexes.
After discussing the fundamentals of the finiteness obstruction, we will
develop some K-theoretic machinery to give a proof of West's theorem.
As an application of West's theorem, we will see that every compact
topological manifold has the homotopy type of a finite CW-complex.
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