Course Info
- Textbook (required) — Hatcher, Allen. Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN: 0-521-79160-X; 0-521-79540-0 MR1867354 (2002k:55001)
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Lecture Notes
- Week 1
- Week 2
- Week 3
- Reading 2 (Due 2/4) - Hatcher p. 134-146)
- Assignment 2 (Due 2/4) - Hatcher p. 155-159 : 2, 4, 11, 14, 17, 19
- 1/26/15 - Lecture 6 — Cellular homology and homology agree, degree, local degree, degree and the cellular boundary map
- 1/28/15 - Lecture 7 — Degree and the cellular boundary map, Free rank of an abelian group, Betti numbers, Euler characteristic, Examples: Sn, Tn, RPn, CPn
- 1/30/15 - Lecture 8 — Hom(-,B) functor, Cochain complexes, Ext functor and properties
- Week 4
- Reading 3 (Due 2/11) - Hatcher p. 146-149 and 190-204
- Assignment 3 (Due 2/11) - Hatcher p. 155-159 : 20, 22, 43 and p. 204-206 : 3, 4, 5
- 2/02/15 - Lecture 9 — Universal coefficient theorem for cohomology, Torsion moves up a dimension for cohomology
- 2/04/15 - Lecture 10 — Tensor product for chain complexes, Künneth formula for chain complexes
- Week 5
- 2/09/15 - Lecture 11 — The cross product and Alexander-Whitney map
- 2/11/15 - Lecture 12 — The Acyclic Models Theorem, Proof of the Eilenberg-Zilber Theorem
- Week 6
- 2/16/15 - Lecture 13 — Proof of the Acyclic Models Theorem
- Assignment 4 (Due 2/18)
- 2/18/15 - Lecture 14 — Applications of the Acyclic Models Theorem, Cross product on cohomology
- 2/20/15 - Lecture 15 — Cup product on cohomology, Cohomology ring
- Week 7
- 2/25/15 - Lecture 17 — Manifolds, Orientation, Existence of fundamental class for orientable manifolds
- 2/27/15 - Lecture 18 — Fundamental class, Slant and cap product
- Week 8
- 3/02/15 - Lecture 19 — Cohomology with compact supports, Relative products, Poincaré Duality for noncompact manifolds
- Reading 5 (Due 3/13) - Hatcher p. 233-257
- Assignment 5 (Due 3/13) - Hatcher p. 257-260 : 2, 3, 7, 8, 10, 11, 12
- Week 10
- Reading 6 (Due 4/1) - Hatcher p. 281-291
- Assignment 6 (Due 4/1) - Hatcher p. 257-260 : 20, 25, 30 and p. 291-292 : 1, 2
- 3/25/15 - Lecture 26 — H-spaces
- 3/27/15 - Lecture 27 — Hopf algebras, Naturality of cross product on cohomology
- Week 11
- Reading 7 (Due 4/8) - Hatcher p. 292-302
- Assignment 7 (Due 4/8) - Hatcher p. 291-292 : 3, 4, 5, 7, 8
- Week 12
- Reading 8 (Due 4/15) - Hatcher p. 303-310
- Assignment 8 (Due 4/15) - Hatcher p. 291-292 : 11, 13, 14
- Week 13
- Reading 9 (Due 4/24) - Hatcher p. 321-327
- Assignment 9 (Due 4/24) - Hatcher p. 310 : 1, 2a-d