Math 612, Algebraic Topology II

Spring 2018

Instructor: Richard Hain

This will be a second semester course in algebraic topology. Topics to be covered include:

Additional topics, such as de Rham's Theorem and the basics of characteristic classes, may also be covered. Students should have taken a standard first semester course in algebraic topology.

Text: Allen Hatcher, Algebraic topology.


  1. Bott, Raoul and Tu, Loring: Differential Forms in Algebraic Topology, Springer GTM 82, 1982.
  2. Greenberg, Marvin J: Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam 1967. (Out of print.)
  3. Hatcher, Allen: Algebraic topology, Cambridge University Press, Cambridge, 2002.
  4. Milnor, John and Stasheff, James: Characteristic Classes, Princeton University Press, 1974.
  5. Munkres, James: Elements of Algebraic Topology, Addison-Wesley, 1984.
  6. Spanier, Edwin H: Algebraic topology, Corrected reprint. Springer-Verlag, New York-Berlin, 1981. (This is a standard reference.)
  7. Weil, Andre: Sur les théorèmes de de Rham, Comment. Math. Helv. 26, (1952). 119--145. (pdf)

Quote: "A topologist is somebody who does not know the difference between a bagel and a coffee cup." Stephen Colbert on donuts and spheres.

Lecture Notes:

  1. Lecture 1 (pdf)
  2. Lecture 2 (pdf)
  3. Lecture 3 (pdf)
  4. Lecture 4 (pdf)
  5. Lecture 5 (pdf)



  • Simplicial complexes (pdf)
  • Notes on topology (pdf)
  • Old problem set on orientations (pdf)
  • The Thom isomorphism (pdf)
  • Poincare duality from Munkres (pdf)
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