# Questions tagged [chromatic-homotopy]

The chromatic-homotopy tag has no usage guidance.

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### How do topological automorphic forms fit into homotopy theory and what makes them interesting?

Topological automorphic forms (TAF) were introduced by Mark Behrens and Tyler Lawson in 2007, being to Shimura varieties what topological modular forms (TMF) is to the moduli stack of elliptic curves.
...

**7**

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**1**answer

401 views

### Last 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 ...

**17**

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310 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 ...

**6**

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**1**answer

300 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 ...

**5**

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**1**answer

155 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 $d_n$...

**5**

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**1**answer

259 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)...

**3**

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243 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 $...

**7**

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129 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 ...

**10**

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150 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//...

**8**

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197 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 ...

**23**

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**2**answers

1k 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 ...

**4**

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173 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. ...

**9**

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350 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 ...

**5**

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139 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 ...

**47**

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**5**answers

9k 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 ...

**19**

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**1**answer

523 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 ...

**4**

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**1**answer

271 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.

**10**

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364 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 &...

**13**

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**1**answer

283 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
$...

**5**

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**1**answer

495 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 ...

**22**

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514 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 ...

**9**

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**1**answer

344 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 {}...

**4**

votes

**1**answer

232 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 ...

**10**

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451 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 ...

**2**

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175 views

### Formal n-buds from BU(n) rather than SU(n)

It's known, from Ravenel's green book, as well as other sources, that we build formal group laws over a ring from n-buds, where an n-bud is essentially a truncated formal group law (sometimes called a ...

**13**

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**1**answer

1k views

### Connection of X(n) spectra to formal group laws

In the proof of the Nilpotence Theorem, or at least in Ravenel's account of it in his Orange Book, a sequence of spectra are used, denoted $X(n)$ with $X(0)=\mathbb{S}$ and and $X(\infty)=MU$ such ...

**2**

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**1**answer

224 views

### Compact MU or BP Modules

Is there a classification of the compact MU or BP modules in any category of spectra? Can the periodicity theorem be finagled to give a MU-module structure on finite spectra?

**4**

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224 views

### Localization at Infinite Wedges of K-theories or BP

This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges ...

**1**

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**1**answer

193 views

### Properties of endmorphism rings of E(n),K(n)-localized spheres

Is it known whether or not the endomorphism rings (or ring spectra) of the localized sphere spectra in $L_nSp$ and $L_{K(n)}Sp$ are Noetherian or not? Are they well understood? Perhaps, in the vein of ...