## 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.