## 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

Course 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.