All Questions
Tagged with chromatic-homotopy homotopy-theory
52 questions
6
votes
0
answers
141
views
Are the $K(n)$-local $E_n$-Adams spectral sequences isomorphic to the Adams-Novikov spectral sequences?
Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed ...
- 155
6
votes
1
answer
822
views
Chromatic homotopy + algebraic geometry =?
In Homotopy Theory there is a famous theorem which shows that every cohomology theory satisfying a certain list of axioms is characterized by a formal group law, and that the spectrum associated to ...
- 2,907
6
votes
0
answers
357
views
On the nilpotence of the attaching maps for $\mathbb C \mathbb P^\infty$
Consider the usual cell structure on $\mathbb C \mathbb P^\infty$. The skeleta are the $\mathbb C \mathbb P^n$’s, and there is one cell in each even degree. So we have cofiber sequences $S^{2n+1} \to \...
- 64k
7
votes
1
answer
425
views
Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
Let $Sp$ be the category of spectra. Let $L : Sp \to Sp_L$ be the localization functor onto a reflective subcategory.
Question 1: Is it ever the case that $L(S^0)$ is not bounded below?
Question 2: ...
- 64k
9
votes
1
answer
328
views
What is the center of Morava $K$-theory?
Let $E$ be an $E_1$ ring spectrum. Then I believe the center of $E$ is an $E_2$ ring spectrum over which $E$ is an $E_1$ algebra, given by the endomorphisms of $E$ as a bimodule over itself.
Question: ...
- 64k
1
vote
0
answers
171
views
A question on $BP$ and $E_\infty$ models for ring spectrums
I am a beginner in this field. My question is
(1) Is the existence of $E_\infty$ ring structure not closed under weak equivalence of ring spectra?
(2) If (1) is true, what is the risk of replacing a ...
- 71
6
votes
1
answer
255
views
Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?
It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
- 64k
3
votes
1
answer
148
views
Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete
Let $E_n$ be the $n$-th Morava $E$-theory and let $K(n)$ denote the $n$-th Morava $K$-theory.
Question: If $M$ is a $K(n)$-local $E_n$-module, then are the homotopy groups $\pi_*(M)$ $L$-complete? (...
- 177
3
votes
0
answers
70
views
Is every finite spectrum $X$ $K(h)$-locally equivalent to a finite spectrum $Y$ with $\dim (K(h)_\ast Y) = \dim ((H\mathbb F_p)_\ast Y)$?
Let $X$ be a finite spectrum and $K = K(h)$ be the $h$th Morava $K$-theory at the prime $p$. Then $\dim_{K_\ast} K_\ast X$ is increasing in $h$, and eventually constant at $\dim H_\ast(X,\mathbb F_p)$....
- 64k
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
4
votes
0
answers
143
views
Which limits commute with all colimits in $T(h)$-local spectra?
Consider the category $Sp_{T(h)}$ of $T(h)$-local spectra. Let $J, K$ be small $\infty$-categories. Recall that $J$-limits said to commute with $K$-colimits in $Sp_{T(h)}$ if, for all functors $F : J \...
- 64k
5
votes
0
answers
525
views
Is there anything special about the Honda formal group?
The "standard" Morava E-theory $E_n$ (at a prime $p$) is typically defined using the so-called "Honda formal group law", the unique FGL $\Gamma_n$ over $\mathbb{F}_{p^n}$ ...
- 1,969
10
votes
1
answer
424
views
Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?
A ring spectrum $E$ is complex oriented if it is equipped with a ring map $MU\rightarrow E$. It is complex orientable if such a ring map exists.
An $E_\infty$-ring $E$ is $E_\infty$-complex oriented ...
- 2,052
3
votes
1
answer
167
views
Can the Picard-graded homotopy of a nonzero object be nilpotent?
Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...
- 64k
5
votes
1
answer
295
views
If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor?
Let $A$ be a homotopy ring spectrum. Then the homology theory $A_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi_\ast A)$. If $A$ is homotopy commutative, then this functor $...
- 64k
4
votes
1
answer
239
views
On the sparsity of the descent spectral sequence computing homotopy groups of the K(n)-local sphere
There is a descent spectral sequence computing $\pi_*L_{K(n)}S^0$ with $E_2$-term
$$E_2^{s,t}\cong H^s_c(\mathbb{G}_n,(E_n)_t)$$
It is mentioned in Barthel-Beaudry (in the description of Figure 3.30) ...
- 155
9
votes
0
answers
405
views
What is the Balmer spectrum of the p-complete stable homotopy category?
When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
- 1,969
5
votes
0
answers
129
views
Solving polynomial equations in $K(h)$-local or $T(h)$-local spectra?
This is the same question as an earlier question of mine, except in a different category.
Let $Spt_{T(h)}^{fin}$ be the category of finite $T(h)$-local spectra. Let $K_0^\oplus(Spt_{T(h)}^{fin})$ be ...
- 64k
6
votes
1
answer
374
views
Does the Lie algebra structure on rational homotopy groups reflect similar information to the formal group structure in characteristic p?
It's well known (c.f. Quillen and Sullivan) that the rational homotopy theory of spaces is equivalent to the homotopy theory of rational DG-algebras; in particular, rational spaces and rational ...
- 1,969
4
votes
2
answers
200
views
Is $\operatorname{dim}_{K(h)_\ast} K(h)_\ast X$ increasing in $h$?
Let $X$ be a finite $p$-local spectrum. For each $h \in \mathbb{N} \cup \{\infty\}$, let $K(h)$ be Morava $K$-theory of height $h$. Recall that the coefficients $K(h)_\ast$ are a graded field, and $K(...
- 64k
4
votes
0
answers
153
views
Are telescopes Noetherian?
Let $p$ be a prime and $h \in \mathbb N$ a height.
Question 1: Does there exist a compact $T(h)$-local spectrum $A$ with a unital multiplication making $\pi_\ast A$ a Noetherian ring?
A priori it's ...
- 64k
12
votes
1
answer
360
views
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 ...
- 2,052
2
votes
0
answers
83
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 ...
- 448
4
votes
1
answer
192
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 ...
- 2,052
8
votes
1
answer
684
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 $E^{\ast}(BU(n))$...
user340793
12
votes
1
answer
703
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 ...
1
vote
1
answer
136
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 &...
- 64k
3
votes
0
answers
109
views
Does $K(n)$ detect minimal $K(n)$-local cell structures?
Let $X$ be a finite spectrum, and let $N = dim_{\mathbb F_p} H_\ast(X;\mathbb F_p)$. I believe that $p$-completion $X^\wedge_p$ may be built as an $N$-cell complex where the cells are shifts of the $p$...
17
votes
2
answers
725
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 ...
- 64k
8
votes
0
answers
232
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 ...
- 901
12
votes
1
answer
283
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 ...
- 2,052
26
votes
1
answer
832
views
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 ...
- 2,052
21
votes
1
answer
2k
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 ...
- 423
7
votes
2
answers
534
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\}...
- 64k
5
votes
0
answers
168
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 ...
- 64k
18
votes
1
answer
840
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 ...
- 11.2k
11
votes
1
answer
636
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 ...
- 1,098
8
votes
2
answers
1k
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 ...
- 2,356
7
votes
0
answers
172
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 ...
- 64k
15
votes
0
answers
313
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: $$X(m, \infty) \to X \to X[0, m].$$ ...
- 6,308
22
votes
2
answers
6k
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)?
23
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 ...
- 899
18
votes
1
answer
594
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 ...
- 4,189
5
votes
1
answer
202
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 $d_n$...
- 899
11
votes
0
answers
206
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//...
- 3,539
11
votes
0
answers
450
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 ...
- 5,770
6
votes
0
answers
157
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 ...
- 18.4k
63
votes
5
answers
18k
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 ...
4
votes
1
answer
361
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.
user62675
26
votes
0
answers
642
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 ...
- 865