This semester in Wuppertal, I am teaching (together with Doosung Park) the follow-up to Jens' Topologie I, Vertiefung Topologie (Topologie II). You can find the time and place on StudieLöwe (MAT170420) and some more information on Moodle. We will start by covering Poincaré duality, and see where we go from there! There will be exercises spread out throughout the lectures (not examinable) and a Mündlicheprüfung at the end of the semester. 

Lecture 1 (April 15): Recollection of cup product on cohomology, showing that S²vS⁴ is not complex project 2-space, relation to homotopy groups of spheres and Hopf's question, definition of an R-orientation. This is found in Hatcher's Algebraic Topology section 3.2. (Exercise: show that all manifolds are F_2-orientable and that a Z-orientable manifold is R-orientable for all rings R).

Lecture 2 (April 20): Recollection of R-orientation and proved a long lemma, whose corollary was that for each compact R-oriented n-manifold M, there is a unique class μ in H_n(M;R) which restricts to the local R-orientation on M. In particular, this shows that RP^{2n+2} is not Z-orientable for any n\geq 0. We briefly saw the definition of the orientation cover. (Exercise: decide if the Klein bottle is orientable or not. Same for S^1).

Lecture 3 (April 22): Recollection of the corollary above, definition of orientation cover of a manifold ending with the statement that if a path connected manifold M has no index 2 subgroups of its fundamental group, then M is Z-orientable. This shows that all spheres (n\geq 2) and all complex projective spaces are Z-orientable. We then defined the cap product and Kronecker pairing between homology and cohomology, wrote down the official statement of Poincaré duality, discussed a relationship between cap and cup products, and then combined everything to compute the cohomology of complex projective space as a graded ring. (Exercise: repeat/adapt/use this proof to compute the R-cohomology of complex and real projective space, finite and infinite, with general coefficients (even Z in the real case!).)

Lecture 4 ( April 27): We discussed cohomology with compact support and proved a generalisation of Poincaré duality to this setting for n-dimensional Euclidean space. At the end, we discussed a little of the history of Poincaré duality. (Exercise: Show that if F:I-->C is a functor and I has a terminal object t, then the colimit of F is just F(t).)

Lecture 5 (April 29): We finally proved Poincaré duality with compact support. First, we discussed the functoriality of the cohomology of a manifold M with compact support in open subsets, which helped us build Mayer--Vietoris arguments, start with the result for Euclidean space from last lecture. (Exercises: Show that filtered colimits [sufficient for us are colimits taken over partially ordered sets] of abelian groups are exact. Show that constant filtered colimits are equivalent to this constant value. Look at the proof in Hatcher's book on pages 246-7 on the fact that the duality map commutes with boundary maps up to a sign.)

Lecture 6 (May 4): The definition of a manifold with boundary was given, we proved that all compact manifolds with boundary have a collar neighbourhood, and we proved a verision of Poincaré duality for compact manifolds with boundary. We also stated Alexander duality, but we will get to then next time.

Lecture 7 (May 6): We proved Alexander duality and the generalisation of the Jordan curve theorem. Then we looked at a bunch of applications of all of these statements, such as various nonembedding theorems and nonboundary theorems.