Algebraic topology I (Math 2410)
Graduate course, Brown University, 2019
Summary
This is a graduate course on algebraic topology, which aims to study topological spaces with algebraic invariants. The spaces of interest will be cell complexes, manifolds, and covering spaces; the primary invariants we will study are the fundamental group and homology. This course is part one of a two-part course.
Textbooks
- Hatcher, Algebraic topology
- Bredon, Topology and geometry
Topic schedule
- Week 1: Hatcher Chapter 0
- Thurs (9/5). Cell complexes, homotopy, deformation retracts
- Week 2: Hatcher Chapter 0, Bredon Section I.14
- Tue (9/10). Cell complex operations, homotopy extension property
- Thurs (9/12). Quotient by contractible/homotopic attachments theorems, homotopy groups
- Week 3: Hacher 1.1 and 1.2, Bredon III.1-2 and III.9
- Mon (9/16). HW1 due
- Tue (9/17). Homotopy constructions, fundamental groups of spheres
- Thurs (9/19). fundamental group of S^1, van Kampen, fundamental theorem of algebra
- Week 4: Hatcher 1.1 and 1.2
- Tue (9/24). van Kampen, application to cell complexes, basepoints, functoriality
- Thurs (9/26). homotopy invariance, van Kampen proof, more examples
- Week 5: Hatcher 1.3, Bredon III.3-6
- Mon (9/30). HW2 due
- Tue (10/1). Covering spaces, path lifting, monodromy
- Thurs (10/3). Covering homotopy, more monodromy
- Week 6: Hatcher 1.3, Bredon III.7-8
- Tue (10/8). Lifting characterization, deck transformations
- Thurs (10/10). Deck transformations, universal covers, covering actions
- Week 7: Hatcher 2.1, Bredon IV.1,2,4
- Mon (10/14). HW3 due (extension to Wednesday)
- Tue (10/15). Classification of covers
- Thurs (10/17). Construction of universal cover, more examples
- Week 8: Hatcher 2.1, Bredon IV.3,5
- Tue (10/22). Homology, homology of delta complexes
- Thurs (10/24). Midterm
- Week 9: Hatcher 2.1, 2.3, Bredon IV.6,16
- Tue (10/29). Singular homology, H0, H_1
- Thurs (10/31). Functoriality, homology of contractible, Hurewciz theorem
- Week 10: Bredon IV.3-6, Hatcher 2.1 and 2.A,
- Tue (11/5). Hurewicz theorem, homotopy invariance
- Thurs (11/7). Homological algebra, relative homology, axioms for homology
- Week 11: Bredon IV.7, Hatcher 2.2
- Tue (11/12). Snake lemma, degree theory
- Thurs (11/14). degree theory, cellular homology
- Week 12: Bredon IV.10, Hatcher 2.2
- Tue (11/19). homology of cell complexes, cellular homology = singular homology
- Thurs (11/21). Cellular homology, Euler characteristic, Mayer-Vietoris
- Week 13: Bredon IV.17-18
- Tue (11/26). Subdivision, Mayer-Vietoris
- Thurs. No class (Thanksgiving).
- Week 14: Bredon IV.17,19
- Tue (12/3). Excision, generalized Jordan curve theorem
- Thurs (12/5). generalized Jordan curve theorem; knot invariants from homology
- Week 15: reading period.