group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
By the construction of complex oriented cohomology theories from formal groups (via the Landweber exact functor theorem), the height filtration of formal groups induces a “chromatic” filtration on complex oriented cohomology theories. Chromatic homotopy theory is the study of stable homotopy theory and specifically of complex oriented cohomology theories by means of and along this chromatic filtration.
More abstractly, this filtering is induced by the prime spectrum of a symmetric monoidal stable (∞,1)-category of the (∞,1)-category of spectra for p-local finite spectra, which is labeled by the Morava K-theories.
More in detail, for each prime $p \in \mathbb{N}$ and for each $n \in \mathbb{N}$ there is a Bousfield localization of spectra
where $K(n)$ is the $n$th Morava K-theory (for the given prime $p$). These arrange into the chromatic tower which for each spectrum $X$ is of the form
The chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the $p$-localization
of $X$.
Since moreover $L_n X$ is the homotopy fiber product (see at smash product theorem and see this remark at fracture square )
it follows that in principle one may study a spectrum $X$ by understanding all its “chromatic pieces” $L_{K(n)} X$.
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
motivation: J-homomorphism and chromatic homotopy
telescopic complexity?, telescopic localization
Original articles include
A quick idea is given in section 6 of
A good historical introduction is in
Comprehensive lecture notes are in
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 (lecture notes)
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres
Brief surveys include
A lightning review of results by Henn with Goerss, Mahowald, Rezk, and Karamanov is in
A collection of various lecture notes and other material is kept at
See also:
Glossary for stable and chromatic honotopy theory (pdf)
David Mehrle, Chromatic homotopy theory: Journey to the frontier, Graduate workshop notes, Boulder 16-17 May 2018, (pdf, pdf)
Discussion of generalization of elliptic cohomology to higher chromatic homotopy theory is discussed in
Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology pdf
Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology II, Homology, Homotopy and Applications, vol. 10(1), 2008, pp.1-36 (pdf, pdf slides)
Relation of chromatic homotopy theory to Goodwillie calculus is discussed in
Greg Arone, Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Inventiones mathematicae, February 1999, Volume 135, Issue 3, pp 743-788 (pdf, doi:10.1007/s002220050300)
Nicholas Kuhn, Goodwillie towers and chromatic homotopy: an overview, Proceedings of the Nishida Fest (Kinosaki 2003), 245–279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007 (arXiv:math/0410342, doi:10.2140/gtm.2007.10.245)
Categorical ultraproducts are used to provide asymptotic approximations in
There are also chromatic approximations in ordinary (not stabilized) homotopy theory:
Aldridge Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), 831-873 (doi:10.1090/S0894-0347-1994-1257059-7, jstor:2152734)
Gijs Heuts, Lie algebras and $v_n$-periodic spaces (arXiv:1803.06325)
Jacob Lurie, Michael Hopkins, Unstable Chromatic Homotopy Theory, lecture notes 2018 (web)