Questions tagged [stable-homotopy-theory]

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Loop-Suspension adjunction unit is stable equivalence

Let $X$ be a sequential spectrum, then the unit of the $(\Sigma,\Omega)$-adjunction yields a map $\eta : X \to \Omega \Sigma X$. The authors of the book Foundations of Stable Homotopy Theory (p. 48) ...
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Simplicial resolutions and the homotopy fixed points spectral sequence

According to this set of notes, which says (paraphrasing): "To construct the homotopy fixed points spectral sequence, we use the fact that the bar construction gives a simplicial resolution of $(...
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1 answer
25 views

Homology of the Eilenberg-MacLane spectrum of Fp with coefficients in Fq for p and q prime

I understand the Steenrod algebra for $\mathbb{F}_{p^n}$ both from classical calculations and a past question, but I'd like to ask about $H\mathbb{F}_{p*} H \mathbb{F}_q$ for $p$ and $q$ different ...
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What is the Steenrod algebra for finite fields?

I understand that the Steenrod algebra for finite fields with $p$ elements ($p$ prime) is understood, but do we know what the Steenrod algebra is for all finite fields? Namely, what is $H\mathbb F_{p^...
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HA, 1.1.1.7, Lurie

In this remark, Lurie states that applying proposition HTT 4.3.2.15 twice, we deduce that $\theta$ is a kan fibration. How is this assertion deduced?
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What does it mean for an inverse limit to be 0?

I am currently working on the infinite-dimensional case of the Atiyah-Hirzebruch spectral sequence where a lot of inverse limits are needed to state useful conditions on convergence. In particular let ...
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  • 21
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1 answer
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Adams' proof of Homotopy Extension Theorem for CW-spectra

I'm studying stable homotopy category, but I find the language of model category hard to understand. For example, I was suggested reading this (which I call "MMSS") Michael Mandell, Peter ...
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1 vote
0 answers
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$\mathbf H R$ is a ring spectrum if $R$ is a ring

The first example of a ring spectrum is probably the Eilenberg-McLane spectrum of a ring $R$. But how is the multiplication $\mu: \mathbf H R \wedge \mathbf H R \to \mathbf H R$ defined? Probably this ...
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What is the essential image of the suspension spectrum functor $\Sigma^\infty$?

Let $\mathsf{hCW}$ denote the homotopy category of CW-complexes and $\mathsf{hCWSpec}$ the homotopy category of CW-spectra (ie. families of CW-complexes $(X_i)_{i\in\mathbb{Z}}$ with connection maps $\...
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2 votes
1 answer
51 views

Excision in the stable homotopy category

Is there a way to make precise the statements (if it holds at all) that excision holds in the stable homotopy category? I am a beginner in this kind of things, and the stable homotopy category for me ...
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Questions about equivalence of Homotopy categories $D(R)$ and $\operatorname{Mod}_{HR}$

Let $R$ a ring and $D(R)$ it's derived category. Two questions: What are homotopy groups of $D(R)$? This terminology is used in this answer. Conjecture: by definition $D(R)$ is obtained from the ...
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Stable cohomology operation and Steenrod algebra

In the Fomenko's Book, we find the characterization of stable cohomology operations $O^S(q,G,H)$ as the projective limit of $$\cdots \longrightarrow \mathcal{H}^{q+n+1}(K(G,n+1);H)\longrightarrow\...
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1 answer
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Suspension map and Stable cohomology operation as inverse limit.

I found in the Fomenko's book that the stable cohomology operation is an inverse limit of $(H^{n+q}(K(G,n);H),f_n)$, where $G,H$ are a group (or rings, or fields for simplicity, it doesn't matter). My ...
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Define homology by simplicial Eilenberg Maclane spectra.

Let $H\mathbb{Z}$ be the Eilenberg Maclane spectrum by $(H\mathbb{Z})^k=\tilde{\mathbb{Z}}[S^k]$. Here $S^1 = \Delta[1]/\partial \Delta[1]$ is the simplicial circle and $S^k = S^1\wedge \cdots \wedge ...
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stable homotopy category Ho(Spectra) of category of spectra

The category of spectra over CW-complexes has as objects sequences $E:= \{E_n \}_{n \in N}$ of CW complexes $E_i$ together with structure maps $S^1 \wedge E_n \to E_{n+1}$. The morphisms $f: E \to F$ ...
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Explicit Description of the map from Adam's $E_2$ term to Continuous Group Cohomology

I am currently working through the paper ''The Homotopy of $L_2V(1)$ for the Prime $3$" by Goerss, Henn, Mahowald which can be found in the book Categorical Decomposition Technique in Algebraic ...
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6 votes
1 answer
132 views

Smash Products of Spectra and Tensor Products of Homotopy Groups

In his book "Stable Homotopy and Generalized Homology" Adams takes a ring spectrum E (let's say E_infinity for simplicity) such that $E_*E$ is flat over $\pi_*E$. He claims that the spectrum ...
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3 votes
1 answer
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Notation clarification in stable homotopy theory

In this paper which surveys the Goodwillie Calculus, the following notation is used in the introduction: $$\pi_*(P_1 I(X)) \cong \pi_*^s(X)$$ where $X$ is a based space, $I$ is the identity functor, ...
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Checking $X \to S[J^{-1}] \wedge X$ is an $S[J^{-1}]$ equivalence.

This is surely a trivial question, since all sources seem to relegate it to an exercise. Here's my attempt at a proof. Is this the idea? Please excuse the wordiness of the exposition, it's from my own ...
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0 votes
1 answer
54 views

E-Equivalences of E-local spectra are weak equivalences

Let $E$ be a spectrum. Following the video linked below, we say that a spectrum $Z$ is $E$-acyclic if $E_*Z=0$ and a spectrum $X$ is $E$-local if $[Z,X]=0$ whenever $Z$ is $E$-acyclic. According to ...
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1 answer
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$E^*(\mathbb{C}P^{\infty})=\bigoplus_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$ or $\prod_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$?

PRELIMINARY DEFINITIONS: Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$ \tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt) $$ So there is a special ...
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4 votes
0 answers
77 views

Are there finitely many trivial stable stems?

One can look up in a table that for example $\pi_4^s = \pi_5^s = 0$. However, it seems to be that the stable homotopy groups of spheres get larger and larger for higher dimensions. Question: is it ...
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1 vote
1 answer
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Degree of a pmap

Today I have started reading stable homotopy. I have came across the notion of a pmap which is basically equivalence class of maps from cofinal sub-spectra. My query is what do we mean by degree of ...
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Stable homotopy groups as a generalized (reduced) homology theory

It is known that $\pi_*^{st}$ defines a generalized homology theory, where $\pi_n^{st}(X) = \text{colim}_{k \geq 0} \pi_{n+k}(\Sigma^k X)$ is the $n$th stable homotopy group of the based space $X$. ...
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0 votes
0 answers
47 views

Representation Theory and Equivariant Stable Homotopy Theory

What is a good book/source to understand Representations of a Group in the sense we use it in Equivariant Stable Homotopy Theory? I've read Barry Simon's book on Representation Theory but would like ...
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3 votes
0 answers
87 views

Which (co)homology theories have (co)chain complexes, spectrum edition?

What homological functors $\mathbf{Sp}\rightarrow\mathbf{Ab}$ arise in the form $H_0\circ T$, for $T:\mathbf{Sp}\rightarrow D(\mathbf{Ab})$ a triangulated functor, and dually for cohomological ...
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3 votes
0 answers
167 views

Mahowald-Hopkins theorem

I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following: Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom ...
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2 votes
1 answer
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Why is the Moore spectrum $S\mathbf{Z}_{(J)}$ a ring spectrum?

Let $J$ be a set of primes and consider the Moore spectrum $S\mathbf{Z}_{(J)}$. In his paper 'The localization of spectra with respect to homology', Bousfield writes that $S\mathbf{Z}_{(J)}$ is a ring ...
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1 vote
0 answers
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A non modern theory of generalized Thom spectra

I've already posted this question on mathoverflow a few days ago and I had no reaction, I hope it is not illegal to repost it on math.stackexchange. I'm new in the subject of stable homotopy theory, ...
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  • 2,089
1 vote
0 answers
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The infinite loop space associated to the spectrum associated to a special $\Gamma$-space

I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim. On the top of ...
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5 votes
0 answers
138 views

$G$-spaces vs spaces with a $G$-action

In equivariant homotopy theory, it seems like one tends to consider "genuine" $G$-spaces or $G$-spectra, rather than spaces (spectra) with a $G$-action. My (rather soft) question is : why is that a ...
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3 votes
0 answers
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Postnikov sections are monoidal functor or not?

For any space/spectrum $X$ one can define the associated tower of Postnikov sections $\{ P^n X\}_{n}$ as a Bousfield localization with respect to all spheres with dimension $>n.$ Therefore, we have ...
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What does it mean for this proposition to hold in $Top_*$?

In chapter 2 (or 1 depending on your edition) of their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim have a certain Proposition 2.1.9 (or 1.1.9) saying that if $A\to X$ is an h-...
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1 vote
0 answers
45 views

Symmetric spectrum associated to a ring

In this paper by Dwyer, Greenlees and Iyengar, we are introduced to symmetric spectra, and more particularly to the notion of an $\mathbb{S}$-algebra. In both the Notation and terminology ($1.5$), ...
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  • 2,980
2 votes
0 answers
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Why the category of CW-complexes is not stable?

I am studying homotopy theory and I would like to understand better what it means for a category to be stable. For instance, the book I'm studying says that "the categories CW∗ and CCh+ have very ...
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1 vote
0 answers
96 views

CW Wedge sum Cofiber sequence

I was reading this paper by Bousfield on the localization of spectra. On page 5, Lemma 1.13, there's a rather small curious technical detail on wedge sum. We have for a limit ordinal $\lambda,B_{\...
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0 votes
1 answer
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How do I prove that all smooth manifolds being homotopic implies contractible?

Let $X$ and $Y$ be smooth manifolds. If all smooth maps from $Y$ to $X$ are homotopic, then show that the identity map on $X$ is homotopic to some constant map(i.e that $X$ is contractible). I have ...
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1 vote
1 answer
139 views

Homology of a spectrum

Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$) The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)...
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4 votes
0 answers
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$KO_*$ groups of $\mathbb{R}P^\infty$, "Snaiths" theorem for $KO$

I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The standard tools to compute these Groups in the complex case rest on the requirement for the cohomology theories $E$...
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  • 127
2 votes
1 answer
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Multiple products on $R^* R$ for a ring spectrum R

Suppose we have a ring spectrum R with multiplication $\mu$ that also has a diagonal map $\Delta$ (for example the sphere spectrum). The cohomology $R^* R$ has a ring structure, like it would for any ...
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3 votes
0 answers
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Notion of stability for Lusternik-Schnirelman category

I was browsing through survey on Lusternik-Schnirelman category and I became curious is it possible to give the definition of the category using a more invariant approach than the classical definition?...
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2 votes
0 answers
36 views

Stable homotopy of exceptional Lie groups

Using Bott periodicity, we know the stable homotopy of all classical compact lie group. So I am wondering if a similar pattern exists for exceptional Lie groups. Unfortunately, I couldn't find a ...
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1 vote
1 answer
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Field spectra and Eilenberg--MacLane spectra?

Apparently thanks to a theorem of Hopkins and Smith, every field spectrum splits into a wedge of Morava K-theories, where we allow the cases $K(0) = H \mathbb{Q}$ and $K(\infty) = H \mathbb{F}_p$. I ...
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2 votes
1 answer
116 views

Confusion about Phantom Maps

A phantom cohomology operation (originally read phantom map) $f: X \rightarrow Y$ is a non-nullhomotopic map such that the induced cohomology operation on the cohomology theories for spaces is trivial....
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  • 10.4k
2 votes
1 answer
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Why is $E^*(X)$ graded commutative?

Context: I'm reading The stable homotopy category by Malkiewich (see his webpage, at Expository writings), and at this point the author hasn't actually introduced spectra, just the properties we would ...
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4 votes
0 answers
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Additivity in algebraic K-theory --- what does it truly mean?

--- Question --- I have seen several definitions of 'additivity' in algebraic K-theory. In all cases, I can more or less see that there is something additive going on. But I have difficulty seeing ...
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2 votes
1 answer
55 views

Spectra and topological diagrams, nlab

I am referring to the long Proposition 1.23 of U. Schreiber's notes in nlab. We let $X$ be a functor from $StdSphere\rightarrow Top^{*/}_{cg}$. He states that there is a map , where $X_i^{seq}=X(S^i)$...
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  • 8,954
2 votes
1 answer
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Uniqueness of a homotopy category.

For a category with weak equivalences $(C,W)$ call $(ho(C),F)$ a homotopy category of $(C,W)$ where $ho(C)$ is a category and $Q \in Fun^{W}(C,ho(C))$ is a functor inverting $W$ if the following ...
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  • 2,066
2 votes
0 answers
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Functoriality of twisted K-theory

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 now. I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, ...
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  • 127
1 vote
0 answers
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Definition of coevaluation map of a stable category defined in Tammo tom Dieck's Algebraic Topology.

The following picture is from [Tammo tom Dieck, Algebraic Topology, pp.176]. Question I want to ask if the statement that $j$ induces an h-equivalence : $$C(V|K)\overset{}{\longrightarrow}C(\mathbb{...
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