Math 549
A solution to the Arf-Kervaire invariant problem
Spring, 2010.
Lecture notes.

This page will give the tablet notes I wrote during each lecture as the semester progresses. It is written in reverse chronological order, so you see the most recent notes first. However this means that helpful links appear when a topic is first mentioned chronologically, meaning the last time it appears below.

Handwritten lecture notes
and links shown in class



Friday April 30, 2010
DCR notes: Completion of the proof of the Reduction Theorem and recap of the course.
The odd primary Kervaire invariant theorem is the subject of Section 6.4 of the green book and of my 1978 paper.
April 26, 2010
DCR notes:
The proof of the Reduction Theorem I
Proofs of some of the assertions made today can be found in Section 4 of the
HHR preprint.
April 28, 2010
DCR notes:
The proof of the Reduction Theorem II
The numbered statements here appear in the
HHR preprint.
Picture of the slice spectral sequence referred to on page 10. It is also shown on page 76 of the preprint.
Thursday April 22, 2010
DCR notes: The formal part of the Slice Theorem.
This is a world premiere performance.
April 19, 2010
DCR notes:
More about the Adams-Novikov spectral sequence
The structure theorems for the Hopf algebroids associated with MU and BP are stated and proved in
Appendix 2 of the green book as A2.1.16 and A2.1.27.
April 21, 2010
DCR notes:
The proof of the Detection Theorem
See my 6th Tokyo lecture.
April 12, 2010
DCR notes:
The Periodicity Theorem for C8
. This material can be found in my Princeton lecture.
April 14, 2010
DCR notes:
Toward the Adams-Novikov spectral sequence
This material can be found in
the green book,
specifically the second section of Chapter 2 and the first section of Appendix 1.
The first two pages of each of these files are blank; keep on clicking.
April 5, 2010
DCR notes:
Toward the slice spectral sequence for MU(4)
The charts of the two spectral sequences can be found towards the end of Hill's Skye handout.
The pictures of the ρ8-slices can be found on page 3 of my Princeton talk.
April 7, 2010
No class
Friday April 2, 2010. DCR notes: The fixed point theorem and some calculations with MU(4).
For the calculations, see the last 3 pages of my third Tokyo lecture.
March 29, 2010
DCR notes:
The slice spectral sequence for MU.
March 31, 2010
DCR notes:
More of the slice spectral sequence for MU and a periodicity theorem.

For a picture of the slice spectral sequence for KR, see the last page of my third Lisbon lecture.
March 22, 2010
DCR notes:
A functorial definition of equivariant spectra
This is a reinterpretation of the definition of Mandell-May.
March 24, 2010
DCR notes:
Setting up the slice spectral sequence
This is similar to my 4th Tokyo lecture.
March 15, 2010
DCR notes: Equivariant spaces and spectra.
See pages 5-8 of my second Lisbon lecture.
For deeper background, see Greenlees-May.
March 17, 2010
DCR notes:
More about equivariant spectra
March 1, 2010
DCR notes: Complex cobordism and formal group laws.
March 3, 2010
DCR notes: Formal group laws and formal A-modules.
Some of this material can be found in Appendix 2 of the green book.
February 22, 2010
DCR notes: More on the Adams spectral sequence and Browder's criterion.
February 24, 2010
DCR notes: Overview of the proof and an introduction to MU-theory
February 15, 2010
DCR notes: Toward the Adams spectral sequence.
February 17, 2010
DCR notes: Introduction to spectra.

Christian Nassau's 210-stem Ext chart
February 8, 2010
No class
February 10, 2010
No class
February 1, 2010
DCR notes: Kervaire-Milnor's use of surgery
to classify exotic spheres.

Hopkins' San Francisco slides
(20 MB file, be patient)
  • Slide 32 shows surgery on the torus.
  • Slide 38 shows the torus with twisted framing.
  • Slide 53 shows Kervaire's example.
February 3, 2010
DCR notes: Review of the Kervaire
invariant problem and
introduction to Steenrod operations.

January 25, 2010
DCR notes: Framed cobordism and
the definition of the Arf invariant.

The Turkish 10 Lira note
January 27, 2010
DCR notes: Kervaire's construction
of a nonsmoothable manifold and
the Kervaire invariant problem.
Holiday January 20, 2010
No notes available
Wikipedia article on the Kervaire invariant

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