As a part of my exam on Algebraic Topology, I have to prepare a brief exposition deepening a topic treated in the course.
The background is:
- basic homotopy theory (fundamental group, theory of covering spaces, Seifert Van Kampen theorem)
- basic homology theory (simplicial and singular homology, the last developed quite in detail, something about homology of spheric complexes, the axioms of Steenrod)
- very basic cohomology theory (here, just the most important definitions and the cup product cohomology ring)
Rather than a particular application or computation (we have done many in class) I would like to deepen some general result of categorical flavour, connecting different parts of the theory.
In this sense the only things that come to my mind are to read somemething about:
- Hurewicz theorem, or other general connections between homotopy and homology.
- Duality between homology and cohomology.
Any suggestion about these two and where to read (concisely) would be most appreciated, as any suggestion about topics I may not aware about at all.
Thanks in advance.