Characteristic Classes Fall 2017

Instructor: Robert Lipshitz

Basic course information

Time: MWF 11:00 - 11:50.
Place: HEDCO Education Building, room 142.
Office hours: By appointment.
Final exam: None.


Math 636 (Algebraic Topology) or instructor's permission.

Recommended textbooks

  • Husemoller, Fiber bundles.
  • Milnor and Stasheff, Characteristic classes.
  • Steenrod, The topology of fibre bundles.
  • Hatcher, Vector bundles and K-theory (book in progress).
  • Bott and Tu, Differential forms in algebraic topology.
  • Conner and Floyd, Differentiable Periodic Maps.
  • Stong, Notes on Cobordism Theory.

Description and goals

Characteristic classes—cohomology classes naturally associated to vector bundles or, more generally, principal bundles—are a key tool in modern {algebraic, differential}×{topology, geometry}. The course starts with an introduction to vector bundles and principal bundles. It then discusses their main characteristic classes—the Euler class, Stiefel-Whitney classes, Chern classes, and Pontrjagin classes. The last part of the class discusses some applications of characteristic classes to bordism. In the process, we will see some nice applications (e.g., to immersions) and review some important parts of algebraic topology (e.g., obstruction theory).


Course requirements

Suggested homework will be given roughly weekly. Students are required to turn in the solution to one homework problem by the middle of the quarter and a second homework problem by the end of the quarter.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Suggested homework


This schedule is tentative, and may change during the quarter.

Week Date Topic References
1 9/25 Course overview. Vector bundles, fiber bundles. [Hu]: §4.1-4.7
[MS]: Ch. 2, 3
[S]: §1-5
[Ha]: pp. 4-27
[BT]: pp. 53-56
  9/27 Fiber bundles with structure, principal G-bundles.  
  9/29 Clutching, G-bundle isomorphism, mixing.  
2 10/2 Equivalence between bundles with structure and principal G-bundles

This is the last day to drop the class without a W.
  10/4 Bundle homotopy lemma and applications. [Hu]: §2.7, 4.8-10, 4.12, 4.13
[MS]: Ch. 5
[S]: §11, 12, 19
[Ha]: pp. 28-31
  10/6 Universal bundles and classifying spaces.  
3 10/9 More on universal bundles / classification. Existence of universal bundles. [Hu]: §4.11, 8.1, 8.2, 8.3, 20.1, 20.2
[MS]: Ch. 5
[Mi56], [Se68]
  10/11 Explicit examples of universal bundles.  
  10/13 Definition of characteristic classes, and interpretation in terms of classifying spaces.  
4 10/16 Thom isomorphism theorem: statement, proof. [Hu]: §18.1-18.7
[MS]: Ch. 9, 10, 11, 12.
  10/18 Euler class.  
  10/20 First properties of the Euler class.  
5 10/23 Obstruction theory and the Euler class.  
  10/25 More Euler class.  
  10/27 (Catching up.)  
6 10/30 Stiefel-Whitney classes and Chern classes: properties. [Ha]: §3.1
[MS]: Ch. 4, 14
[BT]: §20, 21
[Hu]: §17.1-17.6
  11/1 Existence of Stiefel-Whitney and Chern classes.  
  11/3 Splitting principle.  
7 11/6 More on properties and applications of Stiefel-Whitney and Chern classes. [S]: §29-39
[Hu]: §17.9, 17.12, 18.7
[MS]: Ch. 12
  11/8 Cohomology of BO(n). [MS]: Ch. 6, 7
[Hu]: §20.3, 20.5
  11/10 Pontrjagin classes.

The last day to withdraw from the class or change grading option is 11/12.
[MS]: Ch. 15
[BT]: §22, 23
[Hu]: §20.6, 20.7, 20.8
8 11/13 More on cohomology of BSO(n).  
  11/15 More properties of Pontrjagin classes.  
  11/17 More Pontrjagin classes.  
9 11/20 (Catching up)  
  11/22 (Catching up)  
  11/24 Thanksgiving holiday (no class).  
10 11/27 (Catching up)  
  11/29 Bordism. [MS]: Ch. 16, 17, 18
[CF]: Ch. 1
  12/1 Hirzebruch signature theorem and differentiable structures on spheres.

(Last day of classes.)


Papers cited

[Mi56] John Milnor, Construction of universal bundles. II. Ann. of Math. (2) 63 (1956), 430–436.
[Mi56b] John Milnor, On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64 (1956), 399-405.
[Se68] Graeme Segal, Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math. No. 34 1968 105–112.


Handouts will be posted here, in case you lost the physical copy.

  • First Day Handout will be posted here eventually.