Equivariant homotopy theory and Kervaire Invariant One Problem

This is a joint seminar of Independent University of Moscow and Higher School of Economics (Homotpy Seminar of HSE) offered by myself and Grigory Kondyrev under the supervision of prof. Nikita Markaryan in the fall semester of 2016.
The seminar is sheduled on Tuesday, 19:20, room 304 every weak starting from September, 13.

Description

The Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. The question of in which dimensions n there are n-dimensional framed manifolds of nonzero Kervaire invariant is called the Kervaire invariant problem. In 1969 Browder reformulated Kervaire invariant problem in terms of the stable homotopy groups of spheres which translated the problem into the world of stable homotopy theory. In 2016 Hill, Hopkins and Ravenel answered the problem in all the dimensions except n=126. Their solution extensively uses equivariant algebraic topology and chromatic homotopy theory along with some fascinating calculating techniques. In our course we plan to give a survey of equivariant stable homotopy theory and sketch main parts of their proof.

Plan

  1. Overview of the course, the Arf invariant.
  2. * Basics of (00,1)-categories and stable homotopy theory.
  3. From manifolds to spectra: Adams-Novikov spectral sequence and Browder's theorem.
  4. Unstable equivariant homotopy theory, Elmendorf's theorem.
  5. Naive and genuine equivariant spectra, three types of fixed points, equivariant Spanier-Whitehead duality, tom Dieck splitting, Mackey functors, norms.
  6. ** Aside: Atiyah-Segal like completion theorems.
  7. Orientations, cobordisms, Thom spectra, equivariant Thom spectra.
  8. Slice filtration, associated spectral sequence.
  9. * Elements of chromatic theory.
  10. Gap and Slice theorems.
  11. Periodicity and Homotopy Fixed Points theorems.
  12. Detection theorem.
  13. ** Futher directions: Hopkins-Hill's program of parametrized higher category theory.

* May be skipped, if everyone are familiar with the topic.
** If we would have enough time.

Talks

Talk Speaker Date Notes
Introduction and Arf invariant Artem Prikhodko September, 13 Talks 0 and 1
Presentable and stable categories Grigory Kondyrev September, 20 References for the talk: 1, 2, 3
Adams-Novikov spectral sequence Grigory Kondyrev September, 27 Talk 3
Kervaire invariant Nikolai Konovalov October, 4 Talk 4
Browder's theorem Nikolai Konovalov October, 11 Talk 5
Unstable equivariant homotopy theory, Elmendorf's theorem Artem Prikhodko October, 18 Talk 6
Stable equivariant homotopy category, duality theory Artem Prikhodko October, 25 Talk 7
Mackey functors, tom Dieck splitting Artem Prikhodko November, 1
Thom spectra Grigory Kondyrev November, 8 Talk 9
The slice filtration, Gap theorem Artem Prikhodko November, 15 Talk 10
Norms, Homotopy Fixed Points theorem Artem Prikhodko November, 22
Periodicity theorem Artem Prikhodko November, 29 References: Talks 15 and 17 of [9] and section 9 of [6].
Reduction and Slice theorems Nikolai Konovalov December, 6 References: Talk 16 of [9] and sections 6-7 of [6].
Aside: Segal's Burnside ring conjectrure Vladimir Shajdurov December, 13
Detection theorem Grigory Kondyrev December, 20 References: Talk 18 of [9] and section 11 of [6].
Bonus: Parametrized higher category theory Grigory Kondyrev December, 27

References

  1. John Greenlees, Peter May, Equivariant stable homotopy theory.
  2. Peter May, Equivariant homotopy and cohomology theory.
  3. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. Browder's work on the Arf-Kervaire invariant problem (slides).
  4. Matthew Ando, Andrew J. Blumberg, David Gepner, Michael J. Hopkins, Charles Rezk. An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology.
  5. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. The Slice Spectral Sequence for certain $RO(C_{p^n})$-graded Suspensions of $HZ$.
  6. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. On the non-existence of elements of Kervaire invariant one.
  7. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. The Arf-Kervaire Invariant Problem in Algebraic Topology: Introduction.
  8. Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah. Parametrized higher category theory and higher algebra: A general introduction.
  9. MIT Talbot on Equivariant stable homotopy theory and the Kervaire invariant.

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