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 ...
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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 $z_n \in H_n(MO; \mathbb F_2)$...
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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 ...
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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....
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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 ...
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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 ...
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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}^{\...
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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?
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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 ...
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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 $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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, ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 $x\in G$,...
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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 $\...
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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 = ...
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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
$$(TE_n)\wedge S^1\...
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Stable homology operations
Let $x\in (H\mathbb F_2)_n(X)=[S^n,H\mathbb F_2 \wedge X]$ be a homology class for a space $X$. Is there a description of
$$[S^n\overset x\to H\mathbb F_2 \wedge X\overset{Sq^r\wedge id}\to \Sigma^r H\...
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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://...
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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 ...
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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 ...
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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?
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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 ...
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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 $...
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...