Latest results in chromatic homotopy theory
I want to mention five directions where in the last years significant progress has been made in chromatic homotopy theory. This is of course not exclusive!
Unstable chromatic homotopy theory Among the greatest functors in existence is the Bousfield-Kuhn functor $\Phi_n\colon \mathcal{S} \to \mathrm{Sp}$ from spaces to spectra. The composite $\Phi_n\Omega^{\infty}$ agrees with $K(n)$-localization (or $T(n)$-localization, depending on the setup). Thus the $K(n)$-localization of a spectrum just depends on its infinite loop space as a space! But its rather hard to compute with the Bousfield--Kuhn functor. Much work had been done by Mahowald, Davis, Bousfield,... at height 1, but methods for higher heights were lacking. It was a breakthrough when Behrens and Rezk found a way to express it in some cases using topological Andre-Quillen homology.
- Behrens, Rezk: The Bousfield-Kuhn functor and Topological Andre-Quillen cohomology
- Behrens, Rezk: Spectral algebra models of unstable v_n-periodic homotopy theory
A little later, Heuts brought Lie algebras into the game and actually provided a model of ``$v_n$-periodic unstable homotopy'':
- Heuts: Lie algebras and vn-periodic spaces
- Heuts: Lie algebra models for unstable homotopy theory
The chromatic splitting conjecture The chromatic splitting conjecture is about how we can rebuild the sphere from its $K(n)$-localizations. A strong form of this conjecture was found to admit a counterexample at height 2, prime 2 by Beaudry in 2015. A little later, she found out with Goerss and Henn what the picture actually is at height 2, prime 2.
- Beaudry, Goerss, Henn: Chromatic splitting for the K(2)-local sphere at p=2
Higher real K-theories A crucial ingredient in understanding the $K(n)$-local spheres is to understand higher real K-theories, i.e. spectra of the form $E_n^{hG}$ for $G$ a finite subgroup of the Morava stabilizer group. Classically this was mostly feasible for $n=1$ and $n=2$ and a little beyond. A breakthrough was done by Hahn and Shi, who were able to do the computation for all $n$ if $G=C_2$. Their technique was to construct a $C_2$-equivariant map $MU_{\mathbb{R}} \to E_n$.
- Hahn, Shi: Real Orientations of Lubin-Tate Spectra
This technique allowed also to construct $C_{2^n}$-equivariant maps into $E_n$ from norms of $MU_{\mathbb{R}}$ (in full generality, this is due to a recent preprint by Beaudry, Hill, Shi, Zeng) and these norms are computationally partially understood due to the Kervaire-invariant paper and subsequent work. In particular, the homotopy groups of $E_4^{hC_4}$ were computed by Hill, Shi, Wang and Xu.
Chromatic homotopy theory at big primes It is known that chromatic homotopy theory drastically simplifies at "big primes". But exactly how algebraic is it then? A first answer was attempted in the nineties in a brilliant preprint by Franke, which contained a significant gap though. Recently this was very convincingly solved in two quite different ways by Barthel--Schlank--Stapleton and Pstragowski:
- Barthel, Schlank, Stapleton: Chromatic homotopy theory is asymptotically algebraic
- Pstragowski: Chromatic homotopy is algebraic when p>n^2+n+1
The Balmer spectrum for stable equivariant homotopy One of the most fundamental theorems in chromatic homotopy theory is the thick subcategory theorem, which classifies all thick subcategories of finite spectra (according to prime and height). For a time analogous questions for equivariant spectra remained open, but for abelian groups the situation is now completely known and for non-abelian groups there is significant partial information. The papers are:
- Balmer, Sanders: The spectrum of the equivariant stable homotopy category of a finite group
- Barthel, Hausmann, Naumann, Nikolaus, Noel, Stapleton: The Balmer spectrum of the equivariant homotopy category of a finite abelian group
- Barthel, Greenlees, Hausmann: On the Balmer spectrum for compact Lie groups
More could be mentioned. For example, Hausmann's recent breakthrough about equivariant $MU$ and equivariant formal group laws, but his paper has less of a chromatic feel (though it is certainly related). Or the work on nilpotence, beginning with the May conjecture paper by Mathew--Naumann--Noel and extended by Hahn and others -- the May conjecture paper is already from 2014 though... And one could and should mention the series Elliptic Cohomology I - III by Lurie - but much of this was already announced in 2007.
In 2017, the long-standing problem of whether the Brown–Peterson spectrum $\mathrm{BP}$ admits the structure of an $E_\infty$-ring was shown in the negative.
The question dates back to 1975, formulated as Problem 1 of Peter May's “Problems in infinite loop space theory” [May75]:
Problem 1. Does the Brown–Peterson spectrum $\mathrm{BP}$ admits a model as an $E_\infty$-ring spectrum?
This was answered for the prime $2$ in Theorem 5.5.4 of [Law17] and for odd primes in Theorem 1.2 of [Sen17], which we (partly) state below:
Theorem. The $2$-local Brown–Peterson spectrum does not admit the structure of an $E_n$-algebra for any $12\leq n\leq\infty$.
Theorem. Let $p$ be an odd prime. Then the $p$-local Brown–Peterson spectrum does not admit the structure of an $E_{2\left(p^2+2\right)}$-ring .
See also this MathOverflow question.
References
[Law17] “Secondary power operations and the Brown–Peterson spectrum at the prime $2$.” arXiv:1703.00935.
[May75] “Problems in infinite loop space theory”, Conference on homotopy theory (Evanston, Ill., 1974), Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975, pp. 111–125. MR 761724
[Sen17] “The Brown-Peterson spectrum is not $E_{2 (p^2+2)}$ at odd primes.” arXiv:1710.09822.