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. Singular homology of a topological space, Eilenberg–Steenrod axioms
Oct 15
First computations of homology. Brouwer's fixed point theorem and applications.
Nov 5
Homology with coefficients, universal coefficient theorem, cohomology
Nov 12
CW structures and cellular homology. Comparison with singular homology
Nov 19
Cohomology groups of projective spaces, cohomology of cyclic groups, comparison
Nov 26
Proof of the Eilenberg–Steenrod axioms for singular homology
Dec 3
The cross product and the cup product in cohomology
Dec 10
The Künneth theorem
Dec 17
Computation of some cohomology rings and corollaries
