Instructor: Peter Teichner

Lectures: TuTh 11:00-12:30, 85 Evans

Course Control Number: 54970

Prerequisites: 214, 215A

Syllabus: The course will start with a short survey on Morse theory and the s-cobordism theorem. Then the Thom-Pontrjagin construction will be discussed in detail. It translates the group of smooth manifolds (possibly with additional structure) up to bordism into the homotopy groups of certain Thom spaces. This will lead to the notion of Thom spectra and the associated homology theories. In some cases, these can be computed using characteristic classes which form the next topic of the class. We will then use characteristic numbers to write down certain genera (e.g. Todd-genus, L-genus, A-genus, Witten-genus) that have an interpretation as the index of elliptic operators.

Homework: In place of homework, several projects will be proposed throughout the course and participants can choose to work these out in groups of 2-3. There will be a  mini-conference on these topics during the last week of classes.

Discussions on Sept. 9 and 11:  Gluing.pdf  Orientations.pdf

Projects for the mini conference during the last week of classes:

1. 1.Summarize Milnor’s paper On manifolds homeomorphic to the 7-sphere. (David Berlekamp and Yi Liu)
   

2. 2.Describe Nash’s theorem on analytic structures on smooth manifolds.
   

3. 3.Explain the genus one Heegaard decomposition of lens spaces. (Peter Mannisto and Shawn McDougal)
   

4. 4.Show that Kirby’s moves are sufficient to get between two framed surgery descriptions of a given closed, connected, oriented 3-manifold.
   

5. 5.Understand Casson’s example of a simply-connected 4-manifold (with boundary) that has no handle decomposition without 1-handles.
   

6. 6.Explain Whitehead torsion and the s-cobordism theorem.
   

7. 7.Read about Thom’s jet transversality theorem and understand why the set of Morse functions is open and dense.
   

8. 8.Read about Bott periodicity and the relation to tangent bundles of spheres and real projective spaces. (Raicu Claudiu and Yuhao Huang)
   

9. 9.Explain Brown’s representation theorem and apply it to show that every cohomology theory is represented by a spectrum.
   

10. 10. Understand the computation of the Steenrod algebra, i.e. of all stable cohomology operations over Z/2. (Dan Halpern-Leistner and Dan Pomerleano)
    

11. 11. Read about the category of symmetric spectra and understand the monoidal structure.
    

12. 12. Physical motivation of the Witten genus (Dan Berwick-Evans and Arturo Prat-Waldron)