# 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 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 ...

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### Equivariant spectra with coefficients

In “The localization of spectra with respect to homology”, Bousfield describes localizations with respect to Moore Spectra. Given a spectrum $E$, and a group $M$, he describes the spectrum with ...

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### How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2 X$ without splitting an idempotent?

Let $Z$ be an object in a stable (or triangulated/whatever) category $\mathcal C$. I believe it follows from Thomason's theorem (see The classification of triangulated subcategories) that the ...

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### Cobordism cohomology of Lie groups

Are there any results about cobordism cohomology of Lie groups？For example, $\mathrm{MU}^*(\mathrm{SU}(n))$.

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### Group structure on cohomology with coefficients in a spectral 2-type

Let $E$ be a spectrum having exactly two non-trivial homotopy groups, $\pi_k(E)=G$ and $\pi_j(E)=G'$ for $j>k\geq 0$, and having $k$-invariant $\alpha\colon\Sigma^{k}HG\to\Sigma^jHG'$. Also assume ...

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### Is the equivariant Steenrod algebra useful?

I am a newbie to the field, so please excuse any potential obvious gaps in knowledge. I have been wondering of late about the equivariant (dual) Steenrod algebra in the context of genuine $G = C_p$ ...

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### 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 ...

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### Spanier-Whitehead dual of space of natural transformations

Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
...

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### Lattices and stable homotopy groups of spheres

The number $65520$ arises in two very different scenarios:
It occurs in the formula for the theta series of the Leech lattice:
$$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{...

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### Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory

Let $X$ be an scheme. Fix a prime $l$ which is invertible in $X$. Consider the $K(1)$-localization at prime $l$ of algebraic K theory $L_1K(X)$ and $l$-adic completion of etale K theory $K^{et}(X)$.
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### What are the naive fixed points of a non-naive smash product of a spectrum with itself?

Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc.
One ...

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### Which spectra arise from partially ordered commutative monoids?

Thomason showed how any connective spectrum arises from a symmetric monoidal category:
Robert W. Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995), 78–...

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### Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$

My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$.
It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...

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### The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...

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### When is a stable $\infty$-category the stabilization of an $\infty$-topos?

Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...

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### Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?

Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if ...

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### Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?

In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy ...

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### What happened to the last work Gaunce Lewis was doing when he died?

In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...

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### Does a Gysin map depend on the choice of Thom class?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom ...

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### "Phantom" non-equivalences of spectra?

I would like an example of the following situation, or a proof that no such example exists.
$\textbf{Situation}$: Two connective (EDIT: I'm fine with dropping this condition) spectra $X$ and $Y$ such ...

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### $BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$

What is know about the homotopy groups of $S/3$ where $S/3 = \mathrm{hocofib}(S \xrightarrow{\cdot 3} S)$? Otherwise, is there some reference I can consult for the $BP$-ANSS for $S/3$?

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### Homotopy groups of homotopy fixed points of a $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local orthogonal spectrum

Let $G$ be a finite group and $X$ an orthogonal $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local spectrum with an $G$-action that is trivial on $\pi_*X$.
I want to show that then the map $X^{hG}...

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### Unifying the $n$-truncation factorization system in a topos with the $n$-truncation factorization system of a t-structure

(This is a corrected and more detailed version of an earlier question.)
In good circumstances, e.g. the setting of an $\infty$-topos, we have for each $n\geq -1$ an orthogonal factorization system ...

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### Is there a Dold-Kan theorem for circle actions?

There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...

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### When is the category of connective objects a topos?

There is only one stable $\infty$-topos, namely the trivial category. However, the theory of stable $\infty$-categories with $t$-structure is strikingly reminiscent of the theory of topoi, as both ...

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### Monoidal colimit-preserving functor from spaces to $A$-modules

I am reading Lurie's Elliptic Cohomology II and it claims (Section 4.1.3) that for an $\mathbb{E}_\infty$-ring $A$ "there is an essentially unique symmetric monoidal functor $\mathcal{S} \to \...

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### First Pontrjagin class and generator of $\pi_3(\mathrm{SO}(d))$

It is well-known that $H^4(B\mathrm{SO}(d), \mathbb{Z}) \cong \mathbb{Z}$, with a canonical generator given by $p_1$, the first universal Pontrjagin class.
Let's assume $d\geq 5$ so that everything is ...

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### What can I say about an $E_\infty$ ring spectrum with an odd invertible element?

I have an $E_\infty$ ring spectrum $R$. I suspect it is trivial, but I'm not sure. What I do know is that there is an $R$-linear equivalence $R \simeq \Sigma R$. Unless I am very confused, this ...

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### Thom spectra, tmf, and Weierstrass curve Hopf Algebroid

Let $X(4)$ be the Thom spectrum associated to $\Omega SU(4) \to \Omega SU \simeq BU$. Since $X(4)$ is a homotopy commutative ring spectrum, for any spectrum Y we can construct a resolution
$$
Y \wedge ...

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### Geometric fixed points of induction spectrum

I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....

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### Understanding Balmer spectra

$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated ...

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### Is ku reflexive as a spectrum?

Let $S$ and $\rm ku$ denote the sphere spectrum and the connective K-theory spectrum, respectively.
Then is the morphism ${\rm ku} \to F(F({\rm ku},S),S)$ an equivalence?
Here $F$ denotes the mapping ...

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### Multiplicative structures on truncated Moore spectra

As discussed for example in this paper (and this MO thread), there are obstructions for the existence of $A_n$-structures on Moore spectra $M(p^r):=\mathbb{S}/p^r$, which in general don't vanish. In ...

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### (Nonconnective algebraic) K-theory cohomology = K-theory of cohomology?

Situation
Suppose that we have
a commutative ring (or an $E_{\infty}$-ring) $R$ and
a homotopy type $X$.
Then we get a canonical morphism
$$
f \colon K(R ^ {\Sigma ^ \infty X_+}) \to K(R) ^ {\Sigma ^...

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### How to to understand the homology groups $H_*(\Omega_0^\infty S^\infty)$?

The original statement of the Barratt--Priddy theorem says there is an isomorphism of homology groups
$$H_*(\Sigma_\infty)\cong H_*(\Omega_0^\infty S^\infty),$$
where $\Omega_0^\infty S^\infty$ is the ...

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### Truncated Sphere Spectra and their Modules

I'm trying to use truncations $\tau_{\leq n}S$ of the sphere spectrum to ``interpolate'' between $\DeclareMathOperator{\H}{H} \H\mathbb{Z}$ and $S$, and I am struggling to find references for ...

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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{...

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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 ...

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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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### 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}...