MT 808: Algebraic topology

Lecture Notes.

Lecture 1: Overview.

Lecture 2: Homotopy.

Lecture 3: CW and HEP.

Lecture 4: CW complexes and RP^n.

Lecture 5: CP^n; CW pairs have HEP.

Lecture 6: The fundamental group: definition.

Lecture 7: Path-lifting and pi_1(S^1).

Lecture 8: pi_1(S^1), functoriality, and Brouwer's theorems.

Lecture 9: The fundamental theorem of algebra; basepoint dependence.

Lecture 10: Examples and warm-up to Seifert van Kampen.

Lecture 11: Group presentations and Seifert van Kampen.

Lecture 12: Computations of pi_1.

Lectures 13-14: Application to knot theory.

Lecture 15: Covering spaces and pi_1: overview and consequences.

Lecture 16: Covering spaces and pi_1, part two.

Lecture 17: Covering spaces and pi_1, part three.

Lecture 18: Uniqueness of coverings and the universal cover.

Lecture 19: Finishing the classification theorem.

Lecture 20: Regular covers and deck transformations

Lecture 21: Introduction to higher homotopy groups.

Lecture 22: First examples of homotopy groups.

Lecture 23: pi_n(S^n) and fiber bundles.

Lecture 24: The Hopf fibration and exact sequences.

Lecture 25: Simplicial homology.

Lecture 26: Singular homology.

Lecture 27: Zeroth homology and contractible spaces.

Lecture 28: Functoriality and chain homotopies.

Lecture 29: Mayer-Vietoris and Brouwer's theorem, revisited.

Lecture 30: The homology of a surface and the long exact sequence of a pair.

Lecture 31: The Snake Lemma.

Lecture 32: The Subdivision Lemma and excision.

Lecture 33: Simplicial and singular homology agree.

Lecture 34: Eilenberg-Steenrod and cellular homology.

Lecture 35: Change of coefficients and the Euler characteristic.

Lecture 36: The Künneth formula and introduction to cohomology.

Lecture 37: Universal coefficients for cohomology and the cup product.

Lecture 38: The cap product and Poincaré duality.

Lecture 39: The cohomology of projective space and the Borsuk-Ulam theorem.