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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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 ...
nikola karabatic's user avatar
9 votes
1 answer
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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, ...
Doron Grossman-Naples's user avatar
6 votes
3 answers
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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\...
Doron Grossman-Naples's user avatar
1 vote
0 answers
434 views

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,\...
Victor TC's user avatar
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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 ...
Mr. Palomar's user avatar
22 votes
2 answers
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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 ...
D. Zack Garza's user avatar
10 votes
1 answer
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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 ...
John Greenwood's user avatar
7 votes
1 answer
397 views

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,...
Victor TC's user avatar
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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 ...
Harry Gindi's user avatar
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6 votes
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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 $z_n \in H_n(MO; \mathbb F_2)$...
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5 votes
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Topological Hochschild homology using equivariant orthogonal spectra

In the Hesselholt-Madsen paper "On the K-theory of finite algebras over Witt vectors of perfect fields", the authors develop some results concerning the Topological Hochschild homology (THH) of ...
S.S.'s user avatar
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1 answer
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Abelian versions of straightening and unstraightening functors

Let $X$ be a quasi-category (an inner Kan complex), let $\mathfrak{C}(X)$ be its rigidification (its associated simplicial category). J. Lurie in "Higher Topos Theory" proved the following theorem 2.1....
David C's user avatar
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3 votes
1 answer
174 views

Spherical objects and K-theory

My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the ...
Let's user avatar
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5 votes
0 answers
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$\mathbb Z \otimes_\mathbb S \mathbb Z$ is concentrated in degree $0$ : mistake in the argument

I'm not sure this is research level so if this is not appropriate, feel free to move the question to StackExchange. However, I post it here since my "fake proof" is based on a (recent) paper and I'm ...
Maxime Ramzi's user avatar
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7 votes
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A confusion about geometric fixed points via spectral Mackey functors and smashing localisations

Let $G$ be a finite group and $N$ a normal subgroup. One of the modern ways to construct the $\infty$-category of $G$-spectra is as product-preserving spectral presheaves $\text{Sp}^G = \text{Fun}^{\...
user99383532's user avatar
2 votes
0 answers
156 views

Is every simplicial spectrum equivalent to an abelian group simplicial spectrum?

I wonder if every simplicial $S^1$-spectrum stable equivalent to an abelian group simplicial $S^1$-spectrum? It seems that I could use the stable Dold-Kan on the heart and use Postnikov towers?
Nanjun Yang's user avatar
4 votes
0 answers
189 views

Direct image and infinite suspension

I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
Tintin's user avatar
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10 votes
1 answer
442 views

Generalized "Homology Whitehead" -- How much does stabilization remember?

Classically, the (non-local-coefficients) homology Whitehead theorem says that if $X \xrightarrow f Y$ is a map of simple spaces, and if the induced map $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ ...
Tim Campion's user avatar
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1 vote
1 answer
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Dimension of $\ell$-adic Eilenberg-Maclane space

I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell}).$$ In some sense, this is a representable functor, i.e. there exists an $\...
curious math guy's user avatar
1 vote
0 answers
87 views

Extensions of infinite loop spaces

I've been reading Rognes' paper "Algebraic K-theory of 2-adic integers" and I'm confused about something he's doing regarding infinite loop spaces (see Theorem 8.1 of that paper). Let me try to phrase ...
Darmig's user avatar
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2 votes
3 answers
388 views

Definitions of sequential homotopy colimits

Suppose $\mathfrak{M}$ is the category of $S^1$-spectra of simplicial sheaves. I know its sequential homotopy colimits (colimit in $\mathbb{N}$ as usual) coincide with categorial colimits since stable ...
Nanjun Yang's user avatar
11 votes
1 answer
602 views

On the relation between categorification and chromatic redshift

In the introduction to the paper Higher traces, noncommutative motives, and the categorified Chern character, Hoyois, Scherotzke and Sibilla write the following. An important insight emerging from ...
Patriot's user avatar
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5 votes
1 answer
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$KO_*$ groups of $\mathbb{R}P^\infty$, "Snaiths" theorem for $KO$

I posted this question some days ago at math.stackexchange, but didn't receive an answer. I have two questions: I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The ...
Excalibur's user avatar
  • 301
6 votes
2 answers
608 views

moving from sphere spectrum to finite spectrum

I am reading Hatcher's treatment of the Adam's spectral sequence. http://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf On page 20, he states "Thus for each $i$ the groups $\pi_i(Z^k)$ are zero for all ...
Elise's user avatar
  • 225
9 votes
1 answer
484 views

When is Thom isomorphism a ring map?

Let $R$ be an $E_{\infty}$-ring spectrum and $B$ be an $E_\infty$-space. Suppose we have an $E_\infty$-map $$ f: B \to BGL_1(S^0)$$ such that the composite $$f_R: B \to BGL_1(S^0) \to BGL_1(R) $$ is ...
Prasit's user avatar
  • 2,013
10 votes
1 answer
346 views

Homotopy extension of $E_{\infty}$-spaces

Suppose that $X$ is a connected $E_{\infty}$-space, naturally $\Omega X$ is also an $E_{\infty}$-space. Can we classify all $E_{\infty}$-extensions of $X$ by $\Omega X$ (up to homotopy). I mean the ...
Paris's user avatar
  • 707
5 votes
1 answer
484 views

Relation between "triangulated bordism", MO, and $H\mathbb{F}_2$

The unoriented bordism theory $MO$ has a map to $H\mathbb{F}_2$ which is easily described for a space $X$ by pushing forward the fundamental class of a singular manifold to $H_*(X)$. Since $MO$ and $H\...
Connor Malin's user avatar
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11 votes
0 answers
486 views

Chromatic Homotopy Theory and Physics

Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
wonderich's user avatar
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8 votes
0 answers
315 views

Bringing cohomology recipes from algebra to topology?

In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $X$ you want to study. Form a category-with-extra-structure (a site) $\mathcal{C}$ whose objects are, ...
John Greenwood's user avatar
22 votes
3 answers
789 views

Boardman's thesis or mimeographed notes

I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more ...
Lennart Meier's user avatar
7 votes
1 answer
424 views

Cohomology theory with only one Adams operation?

Let $E$ be a multiplicative cohomology theory. Fix a prime p. Call a ring map $\psi^{p}:E\rightarrow E$ an Adams operation if it lifts the Frobenius map $E/p\rightarrow E/p$. It is of course well-...
John Greenwood's user avatar
6 votes
1 answer
699 views

Is there ever a Kunneth isomorphism just for powers?

It's pretty rare for a multiplicative cohomology theory $E$ to have a Kunneth isomorphism $E^\ast(X \times Y) \cong E^\ast(X) \otimes_{E^\ast(pt)} E^\ast(Y)$ for all spaces $X,Y$. Are there any ...
Tim Campion's user avatar
  • 60.5k
3 votes
1 answer
285 views

Infinite loop space of ring spectra: the cup product

I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
Tintin's user avatar
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5 votes
2 answers
495 views

Example of a space X exhibiting the Landweber non-exactness of the additive formal group over the integers?

Landweber exactness gives a criterion for when a complex oriented cohomology theory $E$ can be recovered from the formal group law over $E_{*}$ determined by the complex orientation. That is it gives ...
John Greenwood's user avatar
2 votes
1 answer
199 views

Multiplicity of indecomposable stable summands of $BG^{\wedge}_p$

I am reading the article Homotopy stable classification of $BG^{\wedge}_p$ by Martino-Priddy. Let $P_u$, $P_v$ be $p$-subgroups of a finite group $G$, such that $P_u\leq x^{-1}P_v x$ for some $x\in G$,...
Victor TC's user avatar
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1 vote
1 answer
159 views

Calculus of Functors and Model categories

In Calculus of functors and model categories II Biedermann and Rondigs claim in Corollary 6.18 that the $n$-homogeneous model structure on $\mathrm{Fun}(\mathcal{C}, \mathcal{D})$ is stable if $\...
Niall Taggart's user avatar
3 votes
0 answers
86 views

Lower bounds on "size" of Whitehead covers?

Let $X$ be a nonzero finite spectrum, connective say, and consider the Whitehead tower of $n$-connected covers $\dots \to X\langle n \rangle \to X\langle n-1 \rangle \to \dots \to X\langle 0 \rangle = ...
Tim Campion's user avatar
  • 60.5k
5 votes
0 answers
153 views

Continuous functors, spectra and homology theories

Let $T:\mathbf{Top}_*\to \mathbf{Top}_*$ be a continuous functor and $E$ a spectrum with maps $\sigma_n:E_n\wedge S^1\to E_{n+1}$. We have a new spectrum $TE$ with structure maps $$(TE_n)\wedge S^1\...
FKranhold's user avatar
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2 votes
0 answers
116 views

Stable homology operations

Let $x\in (H\mathbb F_2)_n(X)=[S^n,H\mathbb F_2 \wedge X]$ be a homology class for a space $X$. Is there a description of $$[S^n\overset x\to H\mathbb F_2 \wedge X\overset{Sq^r\wedge id}\to \Sigma^r H\...
syzyg's user avatar
  • 21
9 votes
1 answer
436 views

Universal property of $\mathbb S[z]$ and $E_\infty$-ring maps

Let $\mathbb S[z]$ be the free $E_\infty$-ring spectrum generated by the commutative monoid $\mathbb N$. That is, $\mathbb S[z] = \Sigma^\infty_+ \mathbb N$. In Bhatt-Morrrow-Scholze II (https://...
David Mehrle's user avatar
6 votes
1 answer
771 views

How to construct the Moore spectrum?

I am trying to understand how the Moore spectrum is constructed. And in reading Foundations of Stable Homotopy Theory by David Barnes and Constanze Roitzheim, I see that in example 8.4.7 (pg 340) they ...
Ali Caglayan's user avatar
  • 1,185
7 votes
0 answers
324 views

Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures? While I'm at it: is there an example of a tensor ...
Tim Campion's user avatar
  • 60.5k
14 votes
2 answers
649 views

The $K$-theory homology of the Eilenberg-MacLane spectrum

Let $KU$ be the complex $K$-theory spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. For $n\in \mathbb{Z}$, it is known what the homology groups $KU_{n}(H\mathbb{Z})$ are?
Tsk's user avatar
  • 578
7 votes
0 answers
308 views

Funtoriality of twisted K-theory

I posted this question on math.stackexchange, but received no answer there. In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until ...
Excalibur's user avatar
  • 301
7 votes
1 answer
365 views

Reference request: mod 2 cohomology of periodic KO theory

The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $...
Nicolas Boerger's user avatar
7 votes
1 answer
856 views

Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
user avatar
6 votes
2 answers
809 views

Dold-Kan correspondence in the category of symmetric spectra

The Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes of abelian groups is a classical result. It states that the ...
Surojit Ghosh's user avatar
7 votes
0 answers
330 views

Reference request: complex K-theory as a commutative ring spectrum

Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum? For real $K$-theory I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
David Barnes's user avatar
9 votes
0 answers
302 views

Are there non-obvious finite $E_\infty$ ring spectra?

I see two "obvious" classes of nonzero finite $E_\infty$ ring spectra $R$: $R = \Sigma^\infty_+ (S^1)^{\times n}$ $R = D\Sigma^\infty_+ X$ ($X$ a finite space) Questions: Are there any others? In ...
Tim Campion's user avatar
  • 60.5k
13 votes
1 answer
501 views

How weird can a ring spectrum be if all of its modules are free?

Let $R$ be a ring spectrum. If $\pi_\ast(R)$ is a graded field, then all module spectra over $R$ are free. But I don't believe the converse holds. How badly can it fail? I'm assuming that $R$ is at ...
Tim Campion's user avatar
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