The following syllabus is a rough plan and will possibly change as the course progresses.
A detailed summary of everything we cover 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.
Aug 31
Introduction and overview. Chain complexes of abelian groups and their homology; exact sequences; 5–lemma.
Sept 7
The long exact sequence in homology. Categories and functors.
Sept 14
Natural transformations, products, and coproducts. Modules and the tensor product.
Sept 21
Tensor products and Hom modules, projective and flat modules. Resolutions.
Sept 28
Chain homotopies, fundamental lemma of homological algebra. Derived functors.
Oct 5
Tor and Ext. Eilenberg–Steenrod axioms for homology.
Oct 12
First computations of homology. Brouwer's fixed point theorem and applications.
Nov 2
Singular homology. Homology and cohomology with coefficients.
Nov 9
Universal coefficient theorem. Cell complexes.
Nov 16
Cellular homology. Comparison with singular homology.
Nov 23
Proof of the Eilenberg–Steenrod axioms for singular homology.
Nov 30
Homology of real projective spaces.
Dec 7
The Künneth theorem and cup products.
Dec 14
The Borsuk-Ulam theorem and other applications.
