Questions tagged [chromatic-homotopy]

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Can the Bousfield class of projective space be computed directly?

Recall that the Bousfield class of a spectrum $E$, written $\langle E\rangle$, is the class of spectra $X$ such that $X\wedge E$ is not contractible. For example the Bousfield class of any of the ...
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2 votes
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Map to study $K(n)$-local Picard Group

Let $R$ be an $E_{\infty}$-ring. There's a fiber/cofiber sequence $S$: $gl_1 R \to \text{Pic}(R) \to H(\text{Pic}(R))$, where $\text{Pic}(R) =\pi_0 \text{Pic}(R)$ is the Picard group of $R$. Rotating ...
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4 votes
1 answer
131 views

Are Landweber exact spectra determined by their coefficient ring?

Let $E$ be a Landweber exact ring spectrum. That is, we have a map of homotopy ring spectra $MU\rightarrow E$ and an isomorphism of homology theories $E_*X\simeq MU_*X\otimes_{MU_*}E_*$. Is the ...
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7 votes
1 answer
233 views

$E$-(co)homology of $BU(n)$ (Reference request)

I am currently reading Lurie's notes on Chromatic Homotopy Theory (252x) and in Lecture 4 (https://www.math.ias.edu/~lurie/252xnotes/Lecture4.pdf), he skims through the calculation of $E^{\ast}(BU(n))$...
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458 views

Does the spectrum of Morava E-theory depend only on height?

I almost expect the answer to this question to be no, but I can't find it explicitly said anywhere. Given a formal group law $f$ of height $n$ over a perfect field $k$ of characteristic $p$, we can ...
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1 vote
1 answer
91 views

Can a finite, type $n+k$ spectrum be a (non-iterated) colimit of finite, type $n$ spectra for $k \geq 2$?

By the thick subcategory theorem, if $X, Y$ are finite $p$-local spectra of type $m,n$ respectively, then $Y$ can be built from $Y$ in a finite number of "steps" iff $n \geq m$. Here, a &...
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17 votes
2 answers
665 views

For which $n$ does there exist a closed manifold of (chromatic) type $n$?

Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold which is of type $n$ after $p$-localization? When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R ...
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8 votes
0 answers
164 views

Projective $BP_\ast$-dimension of the $BP$-homology of classifying spaces of finite groups

Fix a prime $p$ and let $G$ be a finite group. Do we know the projective dimension of $BP_\ast (BG)$ as a graded $BP_\ast$-module? Or at least that it is finite? My guess is the following: The ...
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12 votes
1 answer
222 views

Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum?

Suppose $E$ is a complex-oriented ring spectrum whose formal group law is isomorphic to the additive one. As the title suggests, we might as well change the complex orientation so that the formal ...
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Are complex-oriented ring spectra determined by their formal group law?

To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$. Suppose $E$ and $F$ are two complex-oriented ring spectra and ...
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13 votes
1 answer
446 views

Chromatic orientability of manifolds

If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with ...
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22 votes
1 answer
1k views

Why does elliptic cohomology fail to be unique up to contractible choice?

It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
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6 votes
1 answer
186 views

Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum

We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we ...
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726 views

What is so 'coloured' on Chromatic Homotopy Theory

As the title suggest, I would like know the motivation/ historical background why chromatic homotopy theory is called 'chromatic'. Literally, what analogy to colors it might have. Accordings to ...
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9 votes
1 answer
304 views

Applications of equivariant homotopy theory in chromatic homotopy theory

I usually do computations in equivariant homotopy theory, but I would like to learn chromatic homotopy theory where one may use the equivariant techniques, e.g., slice spectral sequences, etc. For ...
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11 votes
0 answers
522 views

Chromatic blueshift and Tate cohomology

Let $R$ be an $L_n$-local ring spectrum. Then one knows that the Tate construction $R^{tC_p}$ (with respect to the trivial $C_p$-action on $R$) is $L_{n-1}$-local; this "blueshift" result is ...
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7 votes
2 answers
468 views

Chromatic t-structures?

Questions: Fix a prime $p$ and $n \in \mathbb N_{\geq 1}$. Does the category $Sp_{K(n)}$ of $K(n)$-local spectra admit a nontrivial $t$-structure? By "nontrivial", I simply mean that $\{0\}...
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5 votes
0 answers
146 views

chromatic minimal cell structures

If $X$ is a finite $p$-local spectrum, then the minimal number of cells needed to construct $X$ is exactly $\dim_{\mathbb F_p} H_\ast(X,\mathbb F_p)$. Is there an analogous result in the $K(n)$-local ...
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18 votes
1 answer
557 views

Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory of a space?

Let $k(n)$ be the $n$th connective Morava K-theory, with $k(n)_* = \mathbb F_p[v]$ where $|v| = 2p^n-2$. If $X$ is a space or a spectrum (assumed bounded below), one can compute $k(n)_*(X)$ using ...
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6 votes
1 answer
339 views

Descent for $K(1)$-local spectra

For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by $$L_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i_K ] {}}} K$$ where $g$ is a topological ...
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  • 111
11 votes
1 answer
512 views

On the relation between categorification and chromatic redshift

In the introduction to the paper Higher traces, noncommutative motives, and the categorified Chern character, Hoyois, Scherotzke and Sibilla write the following. An important insight emerging from ...
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  • 958
7 votes
2 answers
809 views

Why do we study complex orientable cohomology theories

It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology ...
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  • 1,703
11 votes
0 answers
362 views

Chromatic Homotopy Theory and Physics

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 ...
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10 votes
1 answer
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Formal group law for oriented bordism

From this answer I learned that the coefficient ring $MSO^{*}[1/2]$ of oriented bordism with 2 inverted supports an odd formal group law and is infact the universal such ring. Is there a reference/...
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6 votes
0 answers
156 views

Is there an $\infty$-topos of monochromatic spaces?

Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
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12 votes
0 answers
282 views

Does virtual Morava K-theory have an Eilenberg-Moore spectral sequence?

In a recent question, Tim Campion was interested in analyzing the Morava $K$–theory of a space $X$ by dissecting the space into connective and coconnective parts: $$X(m, \infty) \to X \to X[0, m].$$ ...
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20 votes
2 answers
4k views

References and resources for (learning) chromatic homotopy theory and related areas

What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
21 votes
2 answers
2k views

Latest results in chromatic homotopy theory

I started a PhD in chromatic homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where ...
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18 votes
1 answer
486 views

Milnor Conjecture on Lie groups for Morava K-theory

A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...
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  • 3,854
7 votes
1 answer
323 views

Crafting Suspension Spectra

There is a theorem by Hopkins and Smith which states that for every $n > 0$ there is an ideal $I_n = (v_0^{k_0}, \dots, v_n^{k_n})$ such that there exist a spectrum $X_n$ with the following ...
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  • 839
5 votes
1 answer
182 views

Homology of a limit of spectra + Cofiber

I have a countable sequence of finite suspension spectra $X_i$, whose $BP$-homology is a $BP_*(BP)$-comodule. Let's assume $BP_*(X_i) = \Sigma^{d_i} BP_* / (v_0^{k_0}, \dots v_i^{k_i}),$ for some $d_n$...
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5 votes
1 answer
277 views

Map between homology of spectra

Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated ($BP_*(X) = \Sigma^d BP_*/I$, where $I$ has the form $I=(v_0^{i_0}, \dots , v_n^{i_n})$ such that the homology is a $BP_*(BP)...
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  • 839
3 votes
1 answer
258 views

Studying the limit of a sequence of spectra knowing their BP-Homology

QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments. Let $X_n$ be the spectrum such that $...
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9 votes
0 answers
143 views

How to show that a spectrum X is not Chromatically Complete

There are some criteria which tell us when a spectrum $X$ is chromatically complete (it's the homotopy limit of its chromatic tower): It has to be p-local and finite, according to the chromatic ...
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  • 839
11 votes
0 answers
189 views

What are examples of spectra whose mod 2 cohomology contain A//A(n)?

Let $//$ denote the Hopf algebra quotient. We know that: $$HF_{2}^*(ko) \simeq A//A(1)$$ $$HF_2^*(tmf) \simeq A//A(2)$$ By Hopf invariant one, we know there is no $X$ such that $HF_2^*(X) \simeq A//...
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9 votes
0 answers
213 views

Chromatic Completion of Suspension Spectra and affine results

There is the Chromatic Convergence Theorem by Hopkins and Ravanel which states that the homotopy inverse limit of the chromatic tower of a finite spectra $X$ is $X$. Let's call any spectra with this ...
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26 votes
2 answers
2k views

Has anyone seen a nice map of multiplicative cohomology theories?

I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere. I could try to do it myself but I really lack expertise, hence am afraid to miss something or ...
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5 votes
0 answers
199 views

Have chromatic techniques actually been used to compute more stable homotopy groups of spheres?

I have heard the perspective that chromatic homotopy theory is meant to break apart the stable homotopy groups of spheres into manageable pieces, and that this perspective has led to various insights. ...
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11 votes
0 answers
417 views

$E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))$ and Eilenberg-Maclane spaces

$\newcommand{\Z}{\mathbf{Z}}$Let $p$ be a prime. In his answer here, Jacob Lurie conjectured that $E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))\simeq K(\Z/p\Z,n)$ where $E_n$ denotes the ...
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6 votes
0 answers
155 views

Are there versions of highly connected covers of Lie groups with highly periodic homotopy groups?

There is much activity around the study of highly connected covers of Lie groups (well, of their "infinite rank" versions like $\displaystyle{\lim_{N\to\infty}} \ O(N)$, say). Looking at the ...
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58 votes
5 answers
14k views

What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
23 votes
1 answer
780 views

What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...
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4 votes
1 answer
338 views

Localization at the Johnson-Wilson spectrum and rationalization

Is there a clean proof that the $L_n$, localization at $E(n)$, is simply rationalization (i.e. $L_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this here, but I haven't seen an answer.
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10 votes
1 answer
687 views

Morava $K$-theory of $K( \mathbb{Z}/p^2)$

The $p$-adic completion of $K( \mathbb{F}_p)$ is known (by Quillen's calculation) to be $H \mathbb{Z}_p$; in particular, $K(\mathbb{F}_p)$ is acyclic with respect to all Morava $K$-theories $K(n), 0 &...
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13 votes
1 answer
331 views

Completed and uncompleted operations for Morava $E$-theory

Let $E = E_n$ be the $n$-th Morava $E$-theory with coefficient ring $$ E_* = \mathbb{W}(\mathbb{F}_{p^n})[\![u_1,\ldots,u_{n-1}]\!][u^{\pm 1}]. $$ It is usual to consider the completed co-operations $...
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  • 3,525
6 votes
1 answer
656 views

Uniqueness of Complex Orientation of Morava K-theory

It is known that the $n^{\text{th}}$ Morava $K$-theory at a prime $p$, denoted $K(n)$, is complex oriented. In other words, it admits a theory of Chern classes, or equivalently a morphism of homotopy ...
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25 votes
0 answers
600 views

Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect. The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
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9 votes
1 answer
405 views

Morava modules and completed $E$-homology

Let $E = E_n$ be the $n$-th Morava $E$-theory and let $\{ M_{I} \}$ be a tower of generalised Moore spectra. Then (see this previous question) there is a Milnor exact sequence $$0 \to \varprojlim_I {}...
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4 votes
1 answer
260 views

Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?

I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states; For $n\ge2$, the spectra E(n) represent periodic homology theories which at present have ...
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11 votes
0 answers
554 views

Fields in Stable Homotopy Theory

It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...
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