18.917 Infinite families in the homotopy groups of spheres

18.917: Infinite families in the homotopy groups of spheres


Lecturer: Mark Behrens
Time and place: TR 9:30-11:00 Room 2-151
Office: 2-273
E-mail address:

Office hours:

TR 11:00-12:00 (immediately after class)
This class is about infinite families in the unstable and stable homotopy groups of spheres. It will include both a discussion of chromatic (v_n-periodic) families in the stable and unstable stems, and the "Mahowaldean" families, including the Kervaire invariant one problem. Topics will include the Adams and Adams-Novikov spectral sequences, the EHP sequence, the J-homomorphism, and Brown-Gitler spectra. Prerequisites are 18.905-18.906, and (probably) 18.915. I will be assuming basic knowledge of spectra and the Steenrod algebra.


Although I will not be following any textbook in particular, there are some good source materials that you can refer to. The most significant is Ravenel's "Green Book". A great reference for background material for the course is Margolis' book.
  1. D.C. Ravenel: Complex Cobordism and the Stable Homotopy Groups of Spheres ("Green Book"). Online Edition
  2. D.C. Ravenel: Nilpotence and Periodicity in Stable Homotopy Theory ("Orange Book").
  3. H. Margolis: Spectra and the Steenrod Algebra. djvu scan
  4. A.K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257-281.
  5. H. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Alg. 20 (1981) 287-312.
  6. O. Nakamura, On the squaring operations in the May spectral sequence. Mem. Fac. Sci. Kyushu Univ. Ser. A 26 (1972) 293--308.

Topics to be covered (subject to change!)

Lectures (and powerpoint slides, when applicable) will appear next to each of the following topics as they are covered. References (as numbered above) for each topic will also be recorded, e.g. [2],[3]
  1. Overview notes slides
  2. Adams spectral sequences
    1. Construction of ASS's [1] notes
    2. Localizations and completions [2],[4],[5] notes
    3. HF_p: classical ASS and the Steenrod algebra [1],[3] notes
    4. First computations: ku and ko [1] notes slides
    5. Computing Ext: May spectral sequence [1] notes
    6. Power operations and Nakamura's formula [1],[6] notes
    7. Calculation: p=2, low degrees [1] notes slides
    8. Formal groups and BP: brief summary [1],[2] notes
    9. ANSS Calculation: odd prime, low degrees [1] notes slides
  3. Chromatic theory
    1. Chromatic tower and the Chromatic spectral sequence[1][2] notes
    2. Nilpotence, Periodicity, and the monochromatic layers [1][2] notes
    3. Morava change of rings [1] notes
    4. Computing the monochromatic layers: v1-periodicity and the J-spectrum [1] notes
    5. v2-periodicity: the divided beta family [1] notes
    6. EO_n's and the bad primes - the odd primary Kervaire elements [1] notes
  4. Unstable homotopy: the EHP sequence
    1. Construction of the EHP sequence
    2. Comparison with the AHSS for RP^oo
    3. Low dimensional computations
    4. Unstable v_1-periodicity: vector fields on spheres
  5. Mahowaldian families
    1. Hopf invariant 1 and its generalizations
    2. Brown-Gitler spectra
    3. Mahowald's eta_j family
    4. Cohen's zeta_j family
    5. The Kervaire elements