Chromatic homotopy theory explained

In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf.

Chromatic convergence theorem

itself. The theorem was proved by Hopkins and Ravenel.

Let

L_E(n)

denotes the Bousfield localization with respect to the Morava E-theory and let

be a finite,

-local spectrum. Then there is a tower associated to the localizations

… → L_E(2)X → L_E(1)X → L_E(0)X

called the chromatic tower, such that its homotopy limit is homotopic to the original spectrum

The stages in the tower above are often simplifications of the original spectrum. For example,

L_E(0)X

is the rational localization and

L_E(1)X

is the localization with respect to p-local K-theory.

In particular, if the

-local spectrum

is the stable

S_(p)

, then the homotopy limit of this sequence is the original

-local sphere spectrum. This is a key observation for studying stable homotopy groups of spheres using chromatic homotopy theory.

Web site: J. . Lurie . Chromatic Homotopy Theory . 252x (35 lectures) . 2010 . Harvard University.
Web site: J. . Lurie . Unstable Chomatic Homotopy Theory . 2017–2018 . 19 Lectures . Institute for Advanced Study.

http://ncatlab.org/nlab/show/chromatic+homotopy+theory
Web site: Complex Oriented Cohomology Theory and the Language of Stacks . M. . Hopkins . 1999. https://web.archive.org/web/20200620232436/https://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf . 2020-06-20 .
Web site: References, Schedule and Notes . Talbot 2013: Chromatic Homotopy Theory . 2013 . MIT Talbot Workshop.