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Григорий Кондырев, Никита Маркарян, Артем Приходько

Equivariant homotopy theory and the Kervaire Invariant One problem

страница семинара The Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a homotopy 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 2010 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 of the course:

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

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

References:

  1. Edgar H. Brown, Jr. The Kervaire invariant and surgery theory.

  2. John Greenlees, Peter May, Equivariant stable homotopy theory.

  3. 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.

  4. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. The Slice Spectral Sequence for certain $RO(C_{p^n})$-graded Suspensions of $HZ$.

  5. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. On the non-existence of elements of Kervaire invariant one.

  6. Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel. The Arf-Kervaire Invariant Problem in Algebraic Topology: Introduction.

  7. Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah. Parametrized higher category theory and higher algebra: A general introduction.
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