Call number: | 12625 | |
Room/Time: | TuTh 2:40pm-3:55pm, 417 Math | |
Graduate Teaching Assistant: | Song Yu, email: | |
Graduate TA office hours: | Tuesday 11am-2pm in Help Room, 406 Math | |
Instructor: | Mikhail Khovanov | |
Office: | 620 Math | |
Office Hours: | Tentative: Zoom Wednesdays 3-4pm (tentative) and Office (620 Math) Thursdays 1-2pm or by appointment | |
E-mail: | ||
Midterm: | Tuesday, March 7 (tentative) | |
Final exam: | Take-home | |
Webpage: | www.math.columbia.edu/~khovanov/alg_top_2023 | |
Prerequisites: An undergraduate topology course, covering point-set topology and introduction to the fundamental group.
2) A.Fomenko, D.Fuchs, Homotopical topology. Pdf version can be downloaded from the Columbia library website or via Springerlink site for the book. Springer also gives you the option to buy a printed copy for $40, via "MyCopy Softcover" link on the right side of the webpage.
Review of the fundamental group and Seifert--van Kampen theorem. Application to surfaces.
Homotopy and homotopy equivalence. CW complexes. Cellular approximation.
Category theory, functors and adjointness.
Coverings and their classification.
Fibrations and Serre fibrations. Relative homotopy groups.
Complexes and exact sequences. Homotopy sequence of a fibration.
Homotopy groups of CW complexes.
Weak equivalence and cellular approximation. Eilenberg-Maclane spaces.
Homology theory:
Chain complexes and chain maps. Homology of complexes.
Singular homology, homology of CW complexes, computations.
Homotopy and homology, Hurewicz theorem.
Cohomology groups. Homology and cohomology with coefficients.
Kunneth formula. Multiplications in cohomology.
Applications of homology and cohomology.
Manifolds, Poincare duality.
Lecture 5, Tu Jan 31: CW-complexes and Homotopy Extension Property (Hatcher, Chapter 0). Application to homotopy equivalences.