Instructor Info

Name: Nathan Broaddus
Email: broaddus@math.osu.edu
Office: MW (Math Tower) 650
Office Phone: 292 0605

Office Hours

TBA

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)
    Full Text

Lecture Notes

  • Week 1
    • 1/12/15 - Lecture 1 — Axioms for homology
    • 1/14/15 - Lecture 2 — Naturality, tensor products
    • 1/16/15 - Lecture 3 — Tensor products, Tor functor
  • Week 2
    • Reading 1 (Due 1/28) - Hatcher Section 3.A (p. 261-267)
    • Assignment 1 (Due 1/28) - Hatcher p. 267 : 1, 2, 4, 5, 6
    • 1/21/15 - Lecture 4 — Free resolutions, definition of Tor(A,B), Tor(A,B) is well-defined, properties of Tor
    • 1/23/15 - Lecture 5 — Universal Coefficient Theorem, cellular homology
  • 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