The following syllabus is a rough plan and will change and become more detailed as the course progresses.
A summary of everything we have covered so far is available here. This only contains definitions and theorems, no examples, proofs or comments. If you miss a class, you can check there what we've covered and look it up in the literature.
Sept 3
Introduction and overview. Chain complexes of abelian groups and their homology; exact sequences; 5–lemma.
Sept 10
The long exact sequence in homology. Categories and functors.
Sept 17
Natural transformations, products, and coproducts. Modules and the tensor product.
Sept 24
Tensor products and Hom modules, projective and flat modules. Resolutions.
Oct 1
Chain homotopies, fundamental lemma of homological algebra. Derived functors.
Oct 8
Tor and Ext. Eilenberg–Steenrod axioms for homology.
Oct 15
First computations of homology. Brouwer's fixed point theorem and applications.
Nov 5
Singular homology. Homology and cohomology with coefficients.
Nov 12
Universal coefficient theorem. Cell complexes.
Nov 19
Cellular homology. Comparison with singular homology.
Nov 26
Proof of the Eilenberg–Steenrod axioms for singular homology.
Dec 3
Homology of real projective spaces.
Dec 10
The Künneth theorem and cup products.
Dec 17
The Borsuk-Ulam theorem and other applications.
