# 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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### 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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### $E$-(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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### 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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