Syllabus

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:

Amazon logo 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.