Bundles and Characteristic Classes 2021

Instructor: Matt Carr.

Syllabus: As the title of the minicourse suggests, this has essentially two parts.

  • In the first half of the minicourse we will develop some tools about bundles.
  • In the latter half of the minicourse, we will discuss characteristic classes, following Milnor and Stasheff.
  • Time-permitting, we will discuss Chern-Weil theory.

Schedule: We will have two lectures per week, Mondays from 5 p.m. to 6 p.m. and Thursday from 3 p.m. to 4 p.m., starting on June 3rd, in room MW154.

Main reference: Lecture notes (Incomplete! Slowly being updated.)

Extra references: The following are extra references for essentially everything that we will or may talk about.

  • Milnor and Stasheff’s Characteristic Classes.
    • This is the standard reference but uses some antiquated terminology and notation.
  • Switzer’s Algebraic Topology.
    • Switzer’s book is a classic. While it is a little outmoded in its description of spectra and the stable homotopy category, both from the ∞-categorical perspective and the modern symmetric monoidal point-set definitions, it’s still a useful reference. The part relevant to this minicourse is the chapter on characteristic classics and Thom classes. Switzer develops this for a general (complex oriented) cohomology theory.
  • Bott and Tu’s Differential Forms in Algebraic Topology.
    • Chapter IV discusses characteristic classes using de Rham cohomology. Besides this, the book has a bunch of other goodies.
  • Kobayashi and Nomizu’s Foundations of Differential Geometry, Volumes I & II.
    • This is another classic reference. It has a comprehensive treatment of connections in principal G-bundles and the curvature of such connections.
  • Tu’s Differential Geometry: Connections, Curvature, and, Characteristic Classes.
    • This is an interesting book. Tu cam sometimes be imprecise with definitions but when this happens it can be cross-checked. It should be seen as an alternative to Kobayashi and Nomizu’s volumes.
  • Spivak’s A Comprehensive Introduction to Differential Geometry, Volume 2.
    • From chapter 7 onwards, Spivak treats all the various ways of defining connections and shows they are all equivalent.
  • Hirsch’s Differential Topology.
    • This is a good book for understanding smooth manifolds. You should look for the 5th printing. But be aware that others have formed an extensive errata for this book.
  • Kosinski’s Differential Manifolds.
    • This yet another good book for for understanding smooth manifolds. As with the preceding book it must—unfortunately—be read with some caution.

Further references: The following are more technical references that go far beyond what we will discuss.

  • Stong’s Notes on Cobordism Theory.
    • The book doesn’t have a detailed table of contents but Landweber and Ravenel have provided one here. My impression is that the book is really only for those serious about pursuing bordism but the appendix has a nice and careful treatment of some often neglected or incorrectly stated results in differential topology.
  • Hirzebruch’s Manifolds and Modular Forms.
    • I can’t confess to know much about this one. Genera are supposed to be important in theoretical and mathematical physics. Since genera are ring-homomorphisms from cobordism rings, it is unsurprising characteristic classes get involved.