# 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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### 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*}
(-)^\times &\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:
$\mathbb{F}_p\mathrm{Rep}(P,G)\cong \mathbb{F}_p\mathrm{Rep}(P,H)$ 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 ...

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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 ...

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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^{\...

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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 ...

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### Module spectrum maps up to stable homotopy

Let $R$ be a commutative ring spectrum, $M$ and $N$ be a $R$-module spectra.
Let us consider $R$-module maps from $M$ to $N$ up to stable homotopy, that is maps $M \to N$ such that the composites $R \...

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### Every spectrum is the homotopy colimit of shifted suspension spectra

Let $X$ be a spectrum. In various places, I have encountered the statement that
$$
X \simeq \text{hocolim}_n \Sigma^{\infty-n}X_n.
$$
I was wondering how this homotopy colimit is defined, and why we ...

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### When $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$?

When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^...

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### Does the suspension spectrum functor preserve weak equivalences?

Let $\Sigma^{\infty}$ denote the suspension spectrum functor from pointed topological spaces (=CGWH spaces) to orthogonal spectra. As usual, a weak equivalence of spaces is a continuous map inducing a ...

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### Cohomology theories for spaces vs cohomology theories for spectra

It is a standard consequence of the Brown Representability Theorem for $\operatorname{Ho}(\operatorname{Top}_*)$ that the category of generalized cohomology theories for spaces (pointed CW complexes, ...

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### Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category?

In their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $*\leftarrow X\...

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### Bousfield $p$-completion on spectra

Bousfield p-completion on spaces is a functor $(-)^{\wedge p}$ whose main property is that a map $f:X\rightarrow Y$ induces an isomorphism $f_{\ast}:H_\ast(X,\mathbb{F}_{p})\rightarrow H_\ast(Y,\...

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### Two definitions of power operations --- how do they relate?

Below are two different stories about power operations for $\mathbb{E}_\infty$-ring spectra, and I am struggling to see how they relate. In the following we let $R$ be an $\mathbb{E}_\infty$-ring ...

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### What clues originally hinted at stability phenomena in algebraic topology?

If you didn't know anything about stabilization phenomena in algebraic topology and were trying to discover/prove theorems about the homotopy theory of spaces, what clues would point you toward ...

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### Are all classes Stiefel-Whitney classes?

When I thought of this question, I was sure it must have been asked before on this site, but I could't find anything. Maybe my search skills are lacking, or maybe the question is obvious and it's my ...

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### Is $[X, \_]$ a homology theory?

Let $X$ be a CW-spectrum. It is well-known that $[\_ ,X]$ is a generalized cohomology theory and, by Brown's representability theorem, every generalized theory is $H$ represented by a spectrum (namely,...

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### When does QCoh have 'enough perfect complexes'?

Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a ...

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### Generators for unitary bordism ring $\pi_*(MU)$

I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”.
He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$...