## 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:

- singular and simplicial cohomology and their basic properties
- universal coefficient theorems for homology and cohomology
- the Kunneth theorem and cup products
- orientation and duality theorems
- Thom classes, Euler classes and the first Chern class.

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*.

**References:**

- Bott, Raoul and Tu, Loring:
*Differential Forms in Algebraic
Topology*, Springer GTM 82, 1982.
- Greenberg, Marvin J:
*Lectures on algebraic topology*, W. A.
Benjamin, Inc., New York-Amsterdam 1967. (Out of print.)
- Hatcher, Allen:
*Algebraic
topology*, Cambridge University Press, Cambridge, 2002.
- Milnor, John and Stasheff, James:
*Characteristic Classes*,
Princeton University Press, 1974.
- Munkres, James:
*Elements of Algebraic Topology*,
Addison-Wesley, 1984.
- Spanier, Edwin H:
* Algebraic topology*, Corrected reprint.
Springer-Verlag, New York-Berlin, 1981. (This is a standard reference.)
- 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:**

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

**Assignments:**

- Assignment 1: due January 25 (pdf)
- Assignment 2: due February 15 (pdf)
- Assignment 3: due February 22 (pdf)
- Assignment 4: due April 5 (pdf)
- Assignment 5: due April 10 (pdf)
- Assignment 6: due April 17 (pdf)
- Assignment 7: due May 3 (pdf)

**Handouts:**

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

Return to: Richard Hain *
Duke Mathematics Department *
Duke University