Questions tagged [chromatic-homotopy]
The chromatic-homotopy tag has no usage guidance.
55
questions
10
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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 ...
- 1,386
2
votes
0
answers
63
views
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
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1
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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 ...
- 1,386
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1
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$E$-(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 $E^{\ast}(BU(n))$...
- 151
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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 ...
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 &...
- 49.1k
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2
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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 ...
- 49.1k
8
votes
0
answers
164
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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 ...
- 689
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1
answer
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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 ...
- 1,386
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votes
1
answer
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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 ...
- 1,386
13
votes
1
answer
446
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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 ...
- 8,886
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votes
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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 ...
- 433
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1
answer
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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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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
...
- 723
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1
answer
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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 ...
- 745
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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 ...
- 23.3k
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2
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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\}...
- 49.1k
5
votes
0
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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 ...
- 49.1k
18
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1
answer
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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 ...
- 8,886
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1
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339
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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 ...
- 111
11
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1
answer
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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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votes
2
answers
809
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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 ...
- 1,703
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0
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362
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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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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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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 ...
- 49.1k
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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: $$X(m, \infty) \to X \to X[0, m].$$ ...
- 6,008
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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
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2
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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 ...
- 839
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1
answer
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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 ...
- 3,854
7
votes
1
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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 ...
- 839
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1
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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 $d_n$...
- 839
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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)...
- 839
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votes
1
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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 $...
- 839
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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 ...
- 839
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0
answers
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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//...
- 3,379
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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 ...
- 839
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2
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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 ...
- 16.2k
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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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$E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))$ 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 ...
- 4,763
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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 ...
- 16.2k
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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
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1
answer
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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 ...
- 4,640
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1
answer
338
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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.
user62675
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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 &...
- 23.3k
13
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1
answer
331
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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
$...
- 3,525
6
votes
1
answer
656
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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 ...
- 9,157
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0
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600
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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 ...
- 855
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405
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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 {}...
- 3,525
4
votes
1
answer
260
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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 ...
11
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0
answers
554
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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 ...
- 9,157