Algebraic topology II (Math 2420)
Graduate course, Brown University, 2020
Note: This was the first COVID semester, so the course is unusual in some ways.
Summary
This is part-two of a graduate course on algebraic topology. The main topics will be homotopy groups, cohomology, and Poincare duality.
Textbooks
- Hatcher, Algebraic topology
- Bredon, Topology and geometry
Topic schedule
- Week 1: Bredon VII.2-3
- Wed (1/22). Mapping spaces, homotopy classes of maps, loop spaces
- Fri (1/24). H-groups, exponential law
- Week 2: Hatcher 4.1
- Mon (1/27). Suspensions, H-spaces are rare
- Wed (1/29). H-group theorem, homotopy groups
- Fri (1/31). Vanishing homotopy groups of spheres, LES in homotopy
- Week 3: Hatcher 4.1, Bredon VII.6
- Mon (2/3). local PL lemma, LES in homotopy
- Wed (2/5). Fibrations, LES of a fibration
- Fri (2/7). * Fiber bundles
- Week 4: Hatcher 4.2
- Mon (2/10). Serre vs Hurewicz fibrations, LES of fibration
- Wed (2/12). Excision for homotopy groups, Freudenthal suspension
- Fri (2/14). Cellular approximation, homotopy excision
- Week 5: Hatcher 4.2
- Mon (2/17). No class (university holiday).
- Wed (2/19). More homotopy excision, weak homotopy equivalences
- Fri (2/21). Cellular approximation of spaces, w.h.e. is homology isomorphism
- Week 6: Hatcher 4.2
- Mon (2/24). Whitehead’s theorem, proof.
- Wed (2/26). Hurewicz theorem, homotopy excision proof. Midterm released after class.
- Fri (2/28). No class. Midterm due at 5pm.
- Week 7: Bredon II.16, Hatcher 3.1
- Mon (3/2). Pontryagin-Thom theory, cohomology
- Wed (3/4). Cohomology of chain complexes, universal coefficients
- Fri (3/6). Properties of Hom, Ext groups
- Week 8: Hatcher 3.1
- Mon (3/9). Ext groups, universal coefficients theorem
- Wed (3/11). Free resolutions lemma, proof of UCT,
- Fri (3/13). Kunneth theorem, tensor products. Final project proposals due.
Week 9: classes cancelled (COVID-19)
Week 10: Spring break
- Week 11: Bredon VI.1, Hatcher 3.B
- Mon (3/30). Tensor products of abelian groups and chain complexes
- Wed (4/1). Homology cross product, tensor products and Tor
- Fri (4/3). Homology cross product inverse, algebraic Kunneth
- Week 12: Hatcher 3.3, Bredon VI.3-4
- Mon (4/6). Eilenberg-Zilber, algebraic Kunneth theorem proofs
- Wed (4/8). Poincare duality intro, cohomology cross product
- Fri (4/10). Cup products, orientations
- Week 13: Hatcher 3.3
- Mon (4/13). Intuition on cup products and orientations
- Wed (4/15). Cap product, formulations of Poincare duality
- Fri (4/17). Applications of Poincare duality to cobordism
- Week 14: Hatcher 3.3
- Mon (4/20). Fundamental class existence
- Wed (4/22). Cohomology with compact support, Poincare duality proof sketch
- Fri (4/24).
- Week 15: reading period (final project presentations).
- Mon (4/27). Classifying spaces (Li).
- Wed(4/29). Obstruction theory (Dominick+Alex).
- Thurs(4/30). Brown representability (Alina+Veronica).
- Fri(5/1). Cohomology of Eilenberg-Maclane spaces (Zhenghao). Computing π_4(S^3) (Anna)
- Mon(5/4). Computing homology from the nerve of a cover (Yeqiu). Cell structures on manifolds (Owen+Caelan)