65-401:  Algebraic Topology (Master)
Instructor:
Exercise Class: Severin Bunk
Office Hours: see webpage
Content: 1. Homology theory
• Chain complexes
• Singular homology; $$H_0$$ and $$H_1$$; homotopy invariance
• The long exact sequence in homology; the long exact sequence of a pair of spaces
• Excision and the Mayer-Vietoris sequence
• Reduced homology and suspension
• Mapping degree
• CW complexes and Cellular homology
• Homology with coefficients
• Tensor products, the universal coefficient theorem and the topological Künneth formula
2. Singular cohomology
• Definition of singular cohomology
• Universal coefficient theorem for cohomology
• Axiomatic description of a cohomology theory
• Cap product and cup product
• Orientability of manifolds
• Cohomology with compact support and Poincaré duality
• Alexander-Lefschetz duality and applications
• Duality and cup products
• The Milnor sequence
• Lens spaces
Aim: This topology course deals with singular homology and cohomology of topological spaces. Homology groups $$H_n(X)$$, for $$n = 0,1,2...$$ are abelian groups and they are assigned to a space in a functorial way, i.e. for any continuous map $$f\colon X \rightarrow Y$$ there are homomorphisms $$f_*\colon H_n(X) \rightarrow H_n(Y)$$ for $$n=0,1,2....$$ Homology groups are in general easier to calculate than homotopy groups, because they have several structural properties (homotopy invariance, long exact sequences for pairs of spaces, additivity, excision etc). Cellular homology, the Mayer-Vietoris sequence and the Künneth-theorem allow many concrete calculations. On the level of cohomology we have the cup product. This multiplicative structure together with the cap product that combines cohomology and homology, is a further feature that allows us to use algebraic means in order to get geometric statements. We will discuss several examples and some geometric applications such as Poincaré duality.