# 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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### Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres

There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...

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### 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 ...

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### What are some toy models for the stable homotopy groups of spheres?

The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" ...

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### What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category).
Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...

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### Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...

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### What is the Goldie dimension of the ring of stable stems?

Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...

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### The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...

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### Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?

There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...

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### Filtered homotopy colimits of spectra

Let $\mathcal{I}: \mathbb{N} \to \operatorname{Sp}$ be a diagram in the infinity category of spectra. Let $\pi_0(\mathcal{I})$ denote the corresponding $1$-categorical diagram (i.e. compose $\mathcal{...

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### How many automorphisms are there of the category of filtered spectra?

Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly ...

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### An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?

Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...

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### The complex $K$-theory of the Thom spectrum $MU$

The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...

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### Preservation of (co)limits under taking derived categories

Let $R$ be a commutative ring. Let $\{A_i\}_{i \in I}$ be a diagram of $R$-linear $1$-categories, indexed by a finite poset $I$. (If this matters, assume that the $A_i$ have finitely many objects).
...

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### The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction
$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$
descends ...

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### What is the homotopy type of the smash power of Moore spectra $(S/2)^{\otimes n}$?

Let $S/2$ be the mod $2$ Moore spectrum, and let $n \in \mathbb N$.
Question:
What is the homotopy type of the $n$th smash power $(S/2)^{\otimes n}$?
Notes:
When $p$ is odd, we have $S/p \otimes S/p =...

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### What is the colimit of the punctured $k$-cube $\{X^{\epsilon_1 + \dotsb + \epsilon_k} \mid \epsilon_i \in \{0,1\} \text{ not all } 1\}$?

Let $\mathcal C$ be a stably monoidal $\infty$-category, and let $I \xrightarrow f X \to Y$ be a fiber sequence where $I$ is the unit. Then for each $k \in \mathbb N$, we can form a canonical cubical ...

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### Telescopic localisation of Eilenberg-MacLane spaces

Fix a prime $p$ and an integer $n>0$. Let $K$ be the corresponding Morava $K$-theory spectrum, and let $T$ be the telescope of a $v_n$-self map of a finite spectrum of type $n$, and let $X$ be the ...

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### Spectral sequence in Adam's book, Theorem 8.2

I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...

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### Does the homotopy category of finite spectra act on stable homotopy categories?

Assume that C is a stable infinity category; $SH_{fin}$ is the homotopy category of finite spectra. Is there a canonical bi-functor (action? module structure?) $SH_{fin}\times hC \to hC$?
Is there any ...

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### Stable splitting of $\Omega SU(n)$

The space $\Omega SU(n)$ is homotopy-equivalent to $SL_n(\mathbb{C}[z,z^{-1}])/SL_n(\mathbb{C}[z])$. Using this, Steve Mitchell introduced a filtration of $\Omega SU(n)$ by subspaces $F_k$ which can ...

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### What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?

The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...

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### When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups?

Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\...

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### Why the stable module category?

Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...

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### Which spectra have a homotopy-universal connective quotient?

Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...

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### Which spectra have a universal connective quotient?

Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...

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### Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$

Let $\alpha\in H^2(B\operatorname{PSU}(N) ; \mathbb{Z}_N)$ be the obstruction class for lifting a $\operatorname{PSU}(N)$-bundle to an $\mathrm{SU}(N)$-bundle. Note that $\operatorname{PSU}(N)\cong \...

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### Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces

Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...

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### Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$

Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we ...

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### The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)

Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm ...

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### Equivariant complex $K$-theory of a real representation sphere

Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...

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### Equivariant classifying space and manifold models

The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...

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### When can I extend a map of spectra?

Suppose I have a commutative ring $R$. Given an element $(x_1,x_2)\in R^2$ there exists a homomorphism $\mathbb{Z} \to R\otimes R$ taking $1$ to $x_1\otimes x_2$, so there exists a map $f:S^0 \to HR \...

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### Homotopy colimit commutes with homotopy groups

I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem
Let $(E, \mu, \iota)$ be a ring spectrum.
For $\beta \in \pi_n(E)$ an element of the $n$th stable ...

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### Why isn't the anchor map in Lurie's "Rotation Invariance in Algebraic K-Theory" zero?

I think this is a silly question, but I'm quite confused. In Lurie's "Rotation Invariance In Algebraic K-Theory" Notation 3.2.4. he defines a filtered spectrum $\mathbb{A}$ given by
$$\...

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### Homomorphism from Clifford modules to Stable homotopy

In the paper "Clifford modules" by Atiyah, Bott and Shapiro, a homomorphism $\alpha:A_k\rightarrow \tilde{KO}(S^k)$ from a certain group of Clifford modules to real $K$-theory of spheres is ...

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### Worst-case complexity of calculating homotopy groups of spheres

Is the best known worst-case running time for calculating the homotopy groups of spheres $\pi_n(S^k)$ bounded by a finite tower of exponentials? How high is a tower? Does $O(2^{2^{2^{2^{n+k}}}})$ ...

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### Equivariant spectrum with coefficients

I am curious to know whether spectra with coefficients as defined in Adams's Blue book be defined to an equivariant setting. In the non-equivariant case, for a spectrum $E$ and an abelian group $A$, ...

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### Why are ordinary spheres not strictly invertible?

Introduction
This question is about Picard spectra for the symmetric monoidal $\infty$-category of spectra. We say that a spectrum $X$ is invertible if there is another spectrum $Y$ such that $X\...

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### 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 $\mathbb{F}_{p^n}$ ...

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### Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?

In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...

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### Rational G-spectrum and geometric fixed points

For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...

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### About infinite loop space and $\Omega$ spectrum

Let $A$ is an topological abelian monoid. Also $\pi_0(A)$ is a group and $A$ has $CW$ structure.
$BA$ is a classifying space of the topological abelian monoid.
My purpose is to construct an infinite ...

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### infinite families in stable homotopy groups

The question is about infinite families in stable homotopy groups. Yes, there are some Q&A about the topic.
But I wonder if the order of Mahowald's elements is known?
in Green Book it mentioned ...

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### 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 ...

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### If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory?

$\newcommand\Spt{\mathit{Spt}}\newcommand\GrAb{\mathit{GrAb}}$Let $A$ be a ring spectrum. Suppose that $A$ has a Künneth theorem — i.e. the homology theory $A_\ast : \Spt \to \GrAb$ is a strong ...

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### Flatness of objects in a prestable $\infty$-category

I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions?
The typical example is the following. Let $R$ be a connective $\mathbb E_1$-...

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### How to prove that Lie group framing on S^1 represents the Hopf map in framed cobordism

The Pontryagin-Thom construction gives an isomorphism from the stable homotopy groups of spheres and framed cobordism groups. It seems to be well-established that for dimension 1 (see this question), ...

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### Relation between cohomology operations and the Adams spectral sequence

$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Cone}{Cone}$
I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...

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### How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?

In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...

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### $E^G_\ast(E)$ tensored with the rationals

Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...