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Topics in Algebraic Topology: The Sullivan Conjecture >> Content Detail



Syllabus



Syllabus

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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
1Introduction
2Steenrod operations
3Basic properties of Steenrod operations
4The Adem relations
5The Adem relations (cont.)
6Admissible monomials
7Free unstable modules
8A theorem of Gabriel-Kuhn-Popesco
9Injectivity of the cohomology of BV
10Generating analytic functors
11Tensor products and algebras
12Free unstable algebras
13The dual Steenrod algebra
14The Frobenius
15Finiteness conditions
16Some unstable injectives
17Injectivity of tensor products
18Lannes' T-functor
19Properties of T
20The T-functor and unstable algebras
21Free E-infinity algebras
22A pushout square
23The Eilenberg-Moore spectral sequence
24Operations on E-infinity algebras
25T and the cohomology of spaces
26Profinite spaces
27p-adic homotopy theory
28Atomicity
29Atomicity of connected p-Finite spaces
30The Sullivan conjecture
31p-Profinite completion of spaces
32The arithmetic square
33The Sullivan conjecture revisited
34Quaternionic projective space
35Analytic functors revisited
36The Nil-filtration
37The Krull filtration
38Epilogue



Grading


Course grade is based upon class attendance and participation. There are no homework assignments, projects, or exams.


 








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