Thursday seminar Spring 2022
Exponent theorems in homotopy theory
We will study exponent theorems, following the work of Selick, Cohen-Moore-Neisendorfer, Gray, etc., as well as the more recent work of Arone, Heuts, Wang, and others.
Time and place: Thursdays 3:30 - 5:30 pm, in-person in SC 507. (There’ll be a Zoom option, email Mike for a link.)
Tentative schedule:
| Date | Speaker | Topic |
|---|---|---|
| Feb 17 | Mike Hopkins | Introduction and overview of exponent theorems |
| Feb 24 | Ishan Levy; typed notes and handwritten notes | Tools of unstable homotopy theory, James’ 2-primary exponent theorem (Chapters 1-10 of Cohen’s course notes) |
| March 3 | Paul Selick; typed notes | Selick’s theorem on \(S^3\) + Gray’s delooping of the fiber of \(S^{2n-1} \to \Omega^2 S^{2n+1}\) + … |
| March 10 | Robert Burklund; handwritten notes | Starting the proof of CMN: the fiber \(F^{2n+1}\{p^r\}\) of the pinch map \(S^{2n}/p^r \to S^{2n+1}\) |
| March 17 | N/A | Harvard spring break |
| March 24 | Andy Senger | Continuing the proof of CMN: product decomposition of \(\Omega F^{2n+1}\{p^r\}\) and building the map \(\phi_n: \Omega^2 S^{2n+1} \to S^{2n-1}\) |
| March 31 | Paul Selick; typed notes | Complements: Anick spaces and the fiber of the double suspension |
| April 7 | Adela Zhang; handwritten notes | Weiss calculus and derivatives of the identity functor |
| April 14 | Gijs Heuts; handwritten notes | Higher exponents of generalized Moore spaces |
| April 21 | No speaker | Informal discussion session |
Resources:
Papers:
- On the Suspension Sequence by James.
- Torsion in Homotopy Groups and The Double Suspension and Exponents of the Homotopy Groups of Spheres, the original papers of Cohen-Moore-Neisendorfer. Here, they assume \(p\geq 5\). Also see the first three chapters of this book.
- 3-primary exponents by Neisendorfer extends the above results to the prime \(3\).
- Odd primary torsion in \(\pi_k(S^3)\) and A decomposition of \(\pi_\ast(S^{2p+1}; \mathbf{Z}/p\mathbf{Z})\), Selick’s original papers establishing that \(\pi_n(S^3)_{(p)}\) is killed by \(p\) for \(n\geq 4\) and odd \(p\).
- On the iterated suspension by Gray on the fiber of the suspension map \(S^n \to \Omega^r S^{n+r}\).
- EHP spectra and periodicity. I. Geometric constructions by Gray.
- Iterates of the suspension map and Mitchell’s finite spectra with \(A_k\)-free cohomology by Arone.
- Lie algebra models for unstable homotopy theory by Heuts, esp. Section 8(B).
- Unstable chromatic homotopy theory by Wang.
- On the algebras over equivariant little disks by Mike Hill, on a \(C_2\)-equivariant version of the James splitting; also see \(C_2\)-equivariant James splitting and \(C_2\)-EHP sequences by Yigit.
Surveys of some of the older material:
- Algebraic Methods in Unstable Homotopy Theory, a textbook by Neisendorfer. In particular, see Chapter 11.
- A course in some aspects of classical homotopy theory by Cohen gives a fun overview of unstable homotopy theory in the late ’80s.
- A survey of Anick-Gray-Theriault constructions and applications to the exponent theory of spheres and Moore spaces by Neisendorfer.
- Samelson products and exponents of homotopy groups by Neisendorfer.
- Odd primary exponents of Moore spaces by Neisendorfer.
- Periodicity, compositions and EHP sequences by Gray.