Course Meeting Times
Lectures: 3 sessions / week, 1 hour / session
Description
Let G be a finite p-group acting on a nice topological space X. The Sullivan conjecture asserts that the p-adic homotopy type of the fixed point set can be recovered from the action of G on the p-adic completion of the homotopy type of X. The goal of this course is to describe some of the tools (the theory of unstable modules over the Steenrod algebra) which enter into the proof of Sullivan’s conjecture.
Prerequisites
Algebraic Topology II, (18.906). A working knowledge of modern algebraic topology will be assumed, but all of the calculational machinery (such as the Steenrod algebra) will be constructed from scratch.
Text
We will loosely follow the book:
Schwartz, Lionel. Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture. Chicago, IL: University of Chicago Press, 1994. ISBN: 9780226742021.
Calendar
| LEC # | TOPICS |
|---|---|
| 1 | Introduction |
| 2 | Steenrod operations |
| 3 | Basic properties of Steenrod operations |
| 4 | The Adem relations |
| 5 | The Adem relations (cont.) |
| 6 | Admissible monomials |
| 7 | Free unstable modules |
| 8 | A theorem of Gabriel-Kuhn-Popesco |
| 9 | Injectivity of the cohomology of BV |
| 10 | Generating analytic functors |
| 11 | Tensor products and algebras |
| 12 | Free unstable algebras |
| 13 | The dual Steenrod algebra |
| 14 | The Frobenius |
| 15 | Finiteness conditions |
| 16 | Some unstable injectives |
| 17 | Injectivity of tensor products |
| 18 | Lannes’ T-functor |
| 19 | Properties of T |
| 20 | The T-functor and unstable algebras |
| 21 | Free E-infinity algebras |
| 22 | A pushout square |
| 23 | The Eilenberg-Moore spectral sequence |
| 24 | Operations on E-infinity algebras |
| 25 | T and the cohomology of spaces |
| 26 | Profinite spaces |
| 27 | p-adic homotopy theory |
| 28 | Atomicity |
| 29 | Atomicity of connected p-Finite spaces |
| 30 | The Sullivan conjecture |
| 31 | p-Profinite completion of spaces |
| 32 | The arithmetic square |
| 33 | The Sullivan conjecture revisited |
| 34 | Quaternionic projective space |
| 35 | Analytic functors revisited |
| 36 | The Nil-filtration |
| 37 | The Krull filtration |
| 38 | Epilogue |
Grading
Course grade is based upon class attendance and participation. There are no homework assignments, projects, or exams.
