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Questions tagged [stable-homotopy-theory]

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Image of a spectra map to a module spectrum

Let $X$ be a symmetric spectrum and $M$ be an $E$-module for some ring spectrum $E$. Consider a spectra map $f: X \to M.$ Then the image $Im(f)$ may not be an $E$-submodule of $M$ ( even may not be ...
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1answer
50 views

Detecting $\eta^3$ in stunted projective spaces.

Consider the stunted complex projective space $\mathbb{C}P^{n+2}_n:=\mathbb{C}P^{n+2}/\mathbb{C}P^{n-1}$ which is a three-cell complex of the form $$\mathbb{C}P^{n+2}_n\simeq S^{2n}\cup_{\alpha_n} e^...
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1answer
379 views

Connections between Algebraic Topology and Set Theory

(Co) Homology functors are dependent on the homotopy type of the objects they act on and so a lot of results only care about the "loose" classification of spaces (including the use of co-final spectra ...
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0answers
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Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
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1answer
42 views

Mod p cohomology operations and homotopy groups of spheres

It is well-known that "the $p$-torsion in the stable homotopy groups of spheres originate in $\pi_{2p}(S^3)$, but how can I prove this for odd primes? My idea would be to show that the mod $p$-...
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1answer
29 views

Is the complex cobordism spectrum, $MU$, a finite spectrum?

Is the complex cobordism spectrum, $MU$, a finite spectrum? If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite? What about the connective $K$-theory $...
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1answer
41 views

Spectra and long exact sequences

Suppose $X,Y,Z$ are finite CW complexes and let $f \colon X \to Y$ be a fibration with fiber $F$. Is there now a long exact sequence associated to their suspension spectra in the following sense $$ \...
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1answer
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Are there non-contractible space such that suspension $\sigma\colon \pi_k(X) \to \pi_{k+1}(\Sigma X)$ is an isomorphism for all $k$?

Does there exist a path-connected non-contractible CW-complex $X$ such that suspension $\sigma\colon \pi_k(X)\to \pi_{k+1}(\Sigma X)$ is an isomorphism for all $k$? If so is there also a simply-...
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Spectra and cohomology theories

I know that every generalised (Eilenberg-Steenrod) cohomology theory defines a spectrum (in the sense of Lewis-May), and vice-versa. I also know that maps between spectra are richer than maps between ...
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1answer
122 views

Book on stable homotopy theory?

Currently I know nothing about stable homotopy theory other than that it originated from the Freudenthal suspension theorem. But I believe that the following are studied in this field: spectrum, ...
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1answer
50 views

How are multiplication maps of spectra defined?

In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition. For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} \mathrm{MU}(p) \to \...
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Geometric proof of the fact that product of two elements of odd order in image of $J$ is zero.

I want to prove that product of two elements of odd order in image of $J$ is zero. I tried to approach this via the Thom-Pontryagin condition. So the image of $J$ just corresponds to spheres with ...
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0answers
162 views

Where to learn homotopy theory? [closed]

I've discovered recently that homotopy it is more powerful than I thought. I just have some knowledge about classic homotopy theory on topological spaces and simplicial complexes, and very little ...
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1answer
60 views

foundations for the stable homotopy category.

$\newcommand{\C}{\mathscr{C}}$ Where should one learn about the (?) stable homotopy category? I'll call what we're looking for $\C$. There seem to be many competing notions, all of which have some ...
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Structure of module over Eilenberg MacLane spectrum

Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra. Construction: Let $\...
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1answer
47 views

Why is the map from a wedge to a product induces isomorphisms between stable homotopy groups

Let $X_i$ be topological spaces. Why does the map $f:\bigvee_i X_i\rightarrow \prod_i X_i$ induce isomorphisms $\pi^s_*(f)$ on stable homotopy groups?
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39 views

Loops Infinity of a spectrum

Let $\mathbf{X}$ be an (orthogonal) spectrum (can assume that it's an $\Omega$-spectrum if this helps give a positive answer) and give the category of orthogonal spectra the stable model structure. ...
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1answer
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Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively

The question is about understanding Adams spectral sequence intuitively and some of the meanings of its relations. In Adams spectral sequence, $$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
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1answer
38 views

Why Bousfield localization preserves homotopy pull-backs?

Studying chromatic homotopy theory I encountered the chromatic fracture square $\require{AMScd}$ \begin{CD} L_{K(n) \vee K(m)}X @>>> L_{K(m)}X\\ @V V V @VV V\\ L_{K(n)}X @>&...
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3answers
115 views

Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$?

Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$? I can't find any reference about who did it first.
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1answer
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Equality of Hom groups via 5 lemma

Reading a paper I found the following statement: given two spectra $A, B$ since multiplication by $p$ induces the same endomorphism in $[A,B]$ we have $[A \wedge M, B]\cong [A, \Sigma^{-1}M \wedge B]$,...
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0answers
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Why is the $J$-homomorphism an isomorphism for $n=1$?

I am trying to proove that $\pi_{n+1}(S^n) \cong \mathbb{Z}_2$ using the Pontryagin-Thom construction and the special case $n=1$ of the $J$-homomorphism $$ J_1:\pi_1(SO(n))\rightarrow \pi_{n+1}(S^n). $...
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0answers
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Invariant ideals of Hopf algebroid

I have some questions about basic facts of invariant ideals. First the definition: given an Hopf algebroid $(A,\Gamma)$ an ideal $I$ of $A$ is called invariant if $\eta_R(I)=\eta_L(I)$ (see Ravenel, ...
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1answer
54 views

Reference for spectra theory (in topology)

Is there any good reference for the theory of spectra (in topology)? Thanks a lot for your help!
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1answer
61 views

Shift in a stable $\infty$-category

In Proposition 2.1.14 of Lurie's Derived Algebraic Geometry paper (DAG), he gives a triangulated structure on the homotopy category of a stable $\infty$-category. To define the shift operator $A[1]$ ...
3
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1answer
176 views

Is there a stable Hurewicz Theorem?

Given a $(n-1)$-connected spectrum $E$ , is the natural morphism ${\pi _k}\left( E \right) \to {\pi _k}\left( {H\mathbb{Z} \wedge E} \right)$ an isomorphism for $k \leq n$? I think yes, but I can't ...
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0answers
26 views

Localisation with respect to $H \mathbb{Q}$

I am trying to understand what happens when we localise with respect to the cohomology theory with $\mathbb{Q}$ coefficients $H\mathbb{Q}$. In the notes on Morava K theories and localisation by M ...
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0answers
56 views

Equivalence between $SP_{h}(X)$ and $\Gamma^{+}(X)$ in simplicial context

According to the article The homotopy infinite symmetric product represents stable homotopy theory, Proposition 4.5 states: There is an equivalence of topological monoids $$\pi:SP_{h}(X)\...
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0answers
77 views

$\mathrm{\Gamma}$ free group functor of Barratt-Eccles

In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$. Proposition 6.2 states if ...
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Does a map from a compact space to a filtered colimit factor through at a finite stage? [duplicate]

Let $K$ be a compact space, and let $A_i$ be a sequence of spaces in the following diagram. $$A_0 \hookrightarrow A_1 \hookrightarrow A_2 \hookrightarrow \cdots$$ All the inclusions in the diagram are ...
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1answer
126 views

Reference request: Adjunction of $\Sigma^\infty$ and $\Omega^\infty$ is monoidal

I would like to see that the map of spectra $\Sigma^\infty \mathbb{C}P^\infty_+ \to KU$ is actually a map of ring spectra, where $KU$ denotes the complex K-theory spectrum and the map is given as the ...
2
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1answer
71 views

Splitting of a spectrum as a wedge

Suppose that $i:E\to F$ and $r:F\to E$ are maps of spectra ($S^1$-spectra of topological spaces) such that $r\circ i$ is a homotopy equivalence. Can we always show that the spectrum $F$ splits as a ...
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0answers
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Stable Homotopy of Classical Groups

Let $\Gamma$ be the $2n \times 2n$ complex matrix $$ \Gamma = \begin{pmatrix} U & V^* \\ V & U^* \end{pmatrix} $$ where $U$ and $V$ are $n \times n$ complex matrices. Now suppose $\Gamma$ ...
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1answer
179 views

Classifying map of tensor product of two line bundles

We know that $\mathbb{C}P^{\infty}$ is the classifying space of line bundles. Also we know that $\mathbb{C}P^{\infty}$ is an H space that is we have $$\mu: \mathbb{C}P^{\infty} \times \mathbb{C}P^{\...
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0answers
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Defining coefficient ring of a cohomology theory

Let $E$ be a ring spectrum with multiplication given by $\mu$. Then we make $E^*(X)$ a module over $E^*(pt)$ as follows. We give a map $$\phi: [S^{-n},E] \times [\Sigma^{-m}X,E] \to [\Sigma^{-m-n}X,E ]...
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2answers
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Calculating the extraordinary cohomology of $\mathbb{C}P^n$

Let $E$ be a ring spectrum with an orientation. Now I want to calculate $E^*(\mathbb{C}P^n)$. The definition of orientation I am using is: There is an element $x \in E^*(\mathbb{C}P^{\infty})$ such ...
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0answers
49 views

Filtration of the category of finite spectra

Consider the stable homotopy category $\mathcal{SHC}$, we then can define its full subcategory $\mathcal{SHC}^c$ spanned by finite spectra which coincides with the smallest thick triangulated ...
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1answer
71 views

$\pi_0(SO(N))$ and $\pi_0(O(N))$: Inconsistency between Bott periodicity and basic understanding of $\pi_0$

I need to know the homotopy groups of the oriented Grassmannian $\widetilde{Gr}(\infty,\infty) \cong \lim_{N \rightarrow \infty} SO(2N)/(SO(N) \times SO(N))$, and I've run into an inconsistency. It ...
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0answers
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Duality between Thom space and a manifold embedded into a sphere

In a document https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf (s. 19) it is mentioned that there is a map $S^n \to M^+ \wedge \mathrm{Th}\left(\nu \left(M, S^n\right)\right)$, which gives a ...
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1answer
108 views

A spectrum $I$ is $E$-injective iff the map $i:I\rightarrow I\wedge E$ is an inclusion of a retract.

I was reading some notes on stable homotopy theory and I came across the statement in the title of this question. "Suppose $E$ is a ring spectrum, then $I$ is $E$-injective if and only if the ...
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0answers
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Classifying spaces of different finite groups that are stably homotopy equivalent

Can one find non-isomorphic finite groups $G$ and $G'$ such that there is a homotopy equivalence $f:\Sigma^k BG \rightarrow \Sigma^k BG'$ for some $k$? This would be impossible if we assumed $f$ was ...
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stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$ S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2. $$ Question 1: the ...
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2answers
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Why are bordism groups of a point nontrivial

My definitions of Bordism are from Tom Dieck's Algebraic topology book. Briefly, a singular manifold, $M \xrightarrow{f} pt $, for a closed smooth oriented manifold $(M, \omega)$ without boundary is ...
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1answer
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Toy example for computing stable homotopy group.

I am currently reading Hatcher's (Algebraic Topology) explanation of stable homotopy groups. My understanding may be a bit shaky and I made a sort of toy example. I am assuming that $i$ is fixed and ...
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0answers
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Why is Hom$_A(M,A)$ a right $\Gamma$ comodule?

I'm reading through appendix I (Hopf algebroids) of Ravenel's green book, and I came across a line I can't understand in a proof. The part of the lemma I'm interested in states: $\mathbf{Lemma A1.1....
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0answers
64 views

Spectrum of spectrum in the stable homotopy category

Let $\mathcal{E}=(E_0,E_1,\cdots)$ be an $S^1$-spectrum. Define $\Sigma \mathcal{E}$ to be the spectrum with $(\Sigma \mathcal{E})_n=E_{n+1}$. Then, consider the spectrum $\tilde{\mathcal{E}} =(\...
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1answer
51 views

Define a map $\Omega \Sigma \Omega Y \to \Omega Y$

Let $Z$ be $\Omega Y$ which is the space of loops based at $Y_0$. Then I know how to define a map explicitly from $Z \to \Omega \Sigma X$. It is defined by noting we have the identity map $ \Sigma Z,...
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Homotopy fixed points of connective K-theory

Let $ku$ be the $p$-completion of the connective complex K-theory spectrum. The group $\Bbb{Z}_p^\times\cong \Delta \times \Bbb{Z}_p$ acts on $ku$, where $\Delta$ is the cyclic group of order $p-1$. ...
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240 views

When should one learn about $(\infty,1)$-categories?

I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the ...
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1answer
336 views

The ring of stable homotopy groups of spheres is not noetherian

On page 22 of this thesis, it is written that $\pi_*(\Bbb{S})$ is not noetherian. After a bit of thinking and looking online, I haven't found why this is true. A graded ring is noetherian if its ...