Math 612, Algebraic Topology II
Spring 2023
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
- de Rham cohomology and the de Rham theorem
- 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 group homology and cohomology 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.
- Dold, Albrecht: Lectures on algebraic topology. Second edition.
Springer-Verlag, Berlin, 1980.
- 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)
Assignments:
- Assignment 1: due Feb 9 (pdf) (notes)
- Assignment 2: due Feb 28 (pdf)
- Assignment 3: due Mar 9 (pdf)
- Assignment 4: due Mar 21 (pdf)
- Assignment 5: due Apr 4 (pdf)
- Worksheet 1: due Apr 20 (pdf)
- Assignment 6: due Apr 25 (pdf)
- Final Assignment: due 5pm, Thurs, May 4 (pdf)
Handouts:
Notes on simplicial complexes (pdf)
The cohomology ring of a product of spheres (pdf)
Notes from January of 2018 (notes)
Notes on orientations (pdf)
The Thom isomorphism (pdf)
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Duke Mathematics Department *
Duke University