Math 430: Topics in topology: stable homotopy theory

Info:

Office hours:

References

1. H. Margolis: Spectra and the Steenrod Algebra. djvu scan
2. J.F. Adams, Stable Homotopy and Generalised Homology.

Topics to be covered (subject to change!)

1. The stable homotopy category
   1. The Spanier-Whitehead category and Cohomology theories [1],[2] notes
   2. Spectra [1],[2] notes
   3. The stable homotopy category [1],[2] notes
   4. Triangles, smash product, function spectra [1],[2] notes
   5. Generalised homology/cohomology, examples notes
   6. Ring and module spectra, Thom spectra notes
   7. Cellular decomposition, Postnikov systems, Hurewicz theorem notes
   8. Orientation theory,Spanier-Whitehead and Poincare duality notes
   9. Homotopy limits/colimits, homotopy fixed points notes
2. Some spectral sequences
   1. Atiyah-Hirzebruch and Bousfield-Kan spectral sequences notes
   2. Kunneth and universal coefficient spectral sequences notes
3. Cohomology operations notes
   1. Steenrod algebra
   2. Generalised operations and cooperations
   3. Adams operations
4. The Adams spectral sequence
   1. Construction of the ASS notes
   2. Sample computations in the classical ASS notes slides Will Perry's Ext Java Appelet
   3. ANSS
   4. p-adic K-ASS and the J homomorphism
5. Vistas [time permitting]
   1. Modern cateories of spectra (symmetric spectra, etc)
   2. Infinite loop space theory
   3. Algebraic K-theory
   4. Formal groups and MU
   5. Nilpotence and periodicity
   6. Morava E-theories and TMF
   7. Goodwillie Calculus
   8. Equivariant homotopy theory
   9. Motivic homotopy theory