Questions tagged [stable-homotopy]
Stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.
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What is the colimit closure of the finite endomorphism spectra?
$\newcommand{\colim}{\operatorname{colim}}\newcommand{\finend}{\operatorname{finend}}$Let $F$ be a finite spectrum. Then $\operatorname{End}(F) = D(F) \wedge F$ is also finite.
Question: Which spectra ...
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Double coset decomposition for compact Lie groups
The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows
\begin{...
4
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1
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Injectivity of assembly in A-theory for $BC_2 = \mathbb R P^\infty$ in degree $4$
I am trying to understand the assembly map
$$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 ) $$
in low degrees for the space $BC_2 = \mathbb R P^\infty$ in Waldhausen $A$-theory. I know we ...
8
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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 $E^{\ast}(BU(n))$...
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What is the closure of the Eilenberg-MacLane spectra under limits? under colimits?
Every bounded spectrum is in the closure of the Eilenberg MacLane spectra under finite co/limits. Thus every bounded below (resp. above) spectrum is in the closure of the EM spectra under limits (resp....
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A basic computation with spectra
Let $\mathbb{E}=\big(E_n, \sigma_n\colon T\wedge E_n\to E_{n+1}\big)_{n\in\mathbb{N}}$ be a $T$-spectrum, either in the topological setting (with $T=S^1$) or in the algebraic setting (with $T=\mathbb{...
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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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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 $p$...
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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 ...
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Equivariant phantom maps
In the stable homotopy category a map of spectra $f\colon X \rightarrow Y$ is called phantom is the induced map between the associated homology theories $X_* \rightarrow Y_*$ is zero, it is know that ...
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Phantom map in homotopy and homology exists?
Let $f:X\rightarrow Y$ be a map between finite spectrum such that
$\pi_{\ast} (f)=0$. (stable homotopy groups)
$H_{\ast}(f)=0$. (Homology with integer coefficients)
Does it imply that $f$ is null ...
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When is a thick subcategory the preimage of a weak Serre class under a homological functor?
Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $...
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Uniqueness of complex topological $K$-theory as an $S$-algebra
This might be well-known or trivial, but I could not figure out how to fill in the details: For an $S$-algebra $K$ denote its associated multiplicative cohomology theory by $h^*_K$. Suppose that I ...
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Terminology: are there any names for "quotients" of cellular towers in stable categories?
A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; ...
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Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$
We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align*}
\mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\
\mathrm{Inv}...
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A question about cofiber diagrams in stable $\infty$-categories
My question is as follows say I have a commutative diagram
$\require{AMScd}$
\begin{CD}
X @>f>> Y @>g>> Z\\
@V \alpha V V @VV \beta V @VV \gamma V\\
X’ @>>f’> Y @>>g’&...
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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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Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories
There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces):
Symmetric monoidal categories model all connective ...
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Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms
In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...
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Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces
The 1991 paper of Lewis, “Is there a convenient category of spectra?” proved that it is impossible to have a point-set model for spectra satisfying the following criteria:
There is a symmetric ...
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Categorical models for truncations of the sphere spectrum
Picard $n$-groupoids are expected to model stable homotopy $n$-types. So far this has been proved for $n=1$ in
Niles Johnson, Angélica M. Osorno, Modeling stable one-types. Theory Appl. Categ. 26 (...
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Is there essentially unique notion of module over monoidal stable $\infty$-categories?
There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...
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The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126
I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
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Stable Adams operations
I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
8
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Bousfield's distributive lattice DL and non-ring spectra
Bousfield, in his paper "The Boolean algebra of spectra" (Comm Math Helv 54, 368–377 (1979), https://doi.org/10.1007/BF02566281), defined $\mathbf{DL}$, a sublattice of the Bousfield lattice,...
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For which categories $D$ is a $D^{\vartriangleleft\vartriangleright}$-shaped diagram in a stable $\infty$-category a limit iff it is a colimit?
Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and ...
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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 ...
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(Algebraic) cobordism and the rank function
I write the question for algebraic cobordism but I have the analogue question for classic cobordism.
The spectrum representing algebraic cobordism
$$
\mathbf{MGL}=(*, \mathrm{Th}(1) , \ldots , \mathrm{...
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Let $U \in V$ be universes. Is $U-\bf Sp$ always $V-$complete?
Let $\bf Spaces_*$ and $\bf Sp$ denote the categories of pointed spaces and spectra.
Given two functors $F, G: \bf Spaces_* \to Sp$ I would like to define a spectrum of natural transformations. ...
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Stable homotopy groups of complex projective plane
We know that there is a cofiber sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}\mathbb{P}^2$. It's easy to know that $\pi_3^s(\mathbb{C}\mathbb{P}^2)=0$ so there is a surjection
$$\partial:\pi_7^s(S^2\...
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Localizations of spaces with respect to homology and right properness
Let $E$ be a spectrum (with corresponding homology theory denoted $E_\ast$).
In "Localization of spaces with respect to homology", Bousfield constructed a model category structure on the ...
3
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3
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Injectivity of rationalization on spectrum morphisms
Let $E$ and $F$ be two spectra, and let $j \colon F \to F_{\mathbb Q} = F \wedge H \mathbb Q$ be the rationalization of $F$.
Assume that the group of morphisms $[E, F]$ in the stable homotopy category ...
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Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
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Are finite $G$-spectra idempotent complete?
Question: Let $G$ be a compact Lie group (you can assume that $G$ is finite if you like). Is the category of finite $G$-spectra idempotent complete?
Here, by "finite $G$-spectra", I mean ...
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Existence of homotopically non-trivial inclusion map from $X\simeq \mathbb{S}^6$ to $Y\simeq \mathbb{S}^4\vee \mathbb{S}^7$
Let $X$ be a subcomplex of a simplicial complex $Y$ such that $X\simeq \mathbb{S}^6$ and $Y\simeq \mathbb{S}^7\vee\mathbb{S}^4$.
Question: Is the inclusion map $i :X \longrightarrow Y$ always null ...
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How to show that, $ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} $?
Let $ X $ be a $ n $ - dimentional oriented closed real manifold ( i.e : compact and without boundary ).
Can you tell me how to show that,
$$ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq ...
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What is the topological Hochschild cohomology of $\mathbb{F}_p$?
Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}_p$ as outlined in Krause-Nikolaus.
We use the fact that $\mathbb{F}_p$ is initial $E_2$ ring with $0=p$ to compute
...
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Generalization of Hopf invariant
This may be a dumb question, but I ask it here.
In ordinary cohomology, we can construct a Hopf invariant for $f \colon S^{2n-1} \to S^{n}$ by applying $H^{*}(- \colon \mathbb{F}_p)$ to the cofibre ...
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Bigraded endomorphisms of the motivic sphere over a field
In An introduction to $\mathbb A^1$-homotopy theory ([1]) and On the motivic $\pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $\bigoplus_{n\in \mathbb Z} [S^0, \mathbb G_m^{\wedge ...
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Explicit $BP_*BP$-comodule structure on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$
So as it says in the title, how can one explicitly calculate the comodule structures on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$ for a prime $p$?
For example, $\mathbb{C}P^2$ sits in a ...
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What are cospectra, and why have they received so little attention?
A cospectrum (in the context of homotopy theory) is defined to be a sequence of spaces $X_0, X_1, \ldots, X_n, \ldots, $ equipped with maps $X_{n+1}\to \Sigma X_n$, for each $n$. So cospectra are ...
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Solving polynomial equations in spectra?
Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
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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
...
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Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
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$1$-periodic mod-$2$ K-theory
Complex $K$-theory mod $2$ is $2$-periodic, $K/2_* = \mathbf{F}_2[u,u^{-1}]$. Is there an extension $K/2 \to K'$ of ring spectra such that $K'_*=\mathbb{F}_2[q,q^{-1}]$ with $|q|=1$ and such that the ...
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Have mod $p^k$ Dyer Lashof operations been studied?
Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it ...
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Is the forgetful functor $\mathrm{Mod}_R \mathrm{Sp} \rightarrow \mathrm{Sp}$ faithful?
$\DeclareMathOperator{\Sp}{\mathrm{Sp}}$I am taking a special case $\Sp$ here, mainly because it has nice categorical properties.
Let $R$ be an $E_\infty$-ring spectrum. In Higher Algebra, Lurie ...
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Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory?)
The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...
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When does the loop functor $\Omega^\infty:Sp(C) \rightarrow C$ commute with filtered colimits?
Let $C$ be a pointed $\infty$-category which admits finite limits.
Let $Sp(C)$ denote the $\infty$ category of spectrum objects. One way to define, i.e. 1.4.2.24, is by taking the homotopy limit in $...
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$p$-completeness of the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$
Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper A Segal conjecture for $p$-completed classifying spaces, it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\...