Questions tagged [stable-homotopy-theory]
The stable-homotopy-theory tag has no usage guidance.
0
votes
0answers
19 views
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 ...
3
votes
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
$$\mathbb{C}P^{n+2}_n\simeq S^{2n}\cup_{\alpha_n} e^...
1
vote
1answer
379 views
Connections between Algebraic Topology and Set Theory
(Co) Homology functors are dependent on the homotopy type of the objects they act on and so a lot of results only care about the "loose" classification of spaces (including the use of co-final spectra ...
1
vote
0answers
27 views
Tate construction of the Anderson dual of the sphere spectrum
I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
2
votes
1answer
42 views
Mod p cohomology operations and homotopy groups of spheres
It is well-known that "the $p$-torsion in the stable homotopy groups of spheres originate in $\pi_{2p}(S^3)$, but how can I prove this for odd primes?
My idea would be to show that the mod $p$-...
0
votes
1answer
29 views
Is the complex cobordism spectrum, $MU$, a finite spectrum?
Is the complex cobordism spectrum, $MU$, a finite spectrum?
If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite?
What about the connective $K$-theory $...
0
votes
1answer
41 views
Spectra and long exact sequences
Suppose $X,Y,Z$ are finite CW complexes and let $f \colon X \to Y$ be a fibration with fiber $F$. Is there now a long exact sequence associated to their suspension spectra in the following sense
$$
\...
9
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1answer
109 views
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-...
5
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0answers
116 views
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 ...
3
votes
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, ...
1
vote
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 \...
1
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0answers
19 views
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 ...
4
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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 ...
1
vote
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 ...
3
votes
0answers
67 views
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 $\...
0
votes
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?
0
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0answers
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.
...
4
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1answer
70 views
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,
$$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
0
votes
1answer
38 views
Why Bousfield localization preserves homotopy pull-backs?
Studying chromatic homotopy theory I encountered the chromatic fracture square
$\require{AMScd}$
\begin{CD}
L_{K(n) \vee K(m)}X @>>> L_{K(m)}X\\
@V V V @VV V\\
L_{K(n)}X @>&...
2
votes
3answers
115 views
Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$?
Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$? I can't find any reference about who did it first.
6
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1answer
55 views
Equality of Hom groups via 5 lemma
Reading a paper I found the following statement: given two spectra $A, B$ since multiplication by $p$ induces the same endomorphism in $[A,B]$ we have $[A \wedge M, B]\cong [A, \Sigma^{-1}M \wedge B]$,...
2
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0answers
86 views
Why is the $J$-homomorphism an isomorphism for $n=1$?
I am trying to proove that $\pi_{n+1}(S^n) \cong \mathbb{Z}_2$ using the Pontryagin-Thom construction and the special case $n=1$ of the $J$-homomorphism
$$
J_1:\pi_1(SO(n))\rightarrow \pi_{n+1}(S^n).
$...
1
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0answers
26 views
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, ...
3
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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!
1
vote
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
votes
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 ...
1
vote
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 ...
0
votes
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 $$\pi:SP_{h}(X)\...
3
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0answers
77 views
$\mathrm{\Gamma}$ free group functor of Barratt-Eccles
In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$.
Proposition 6.2 states
if ...
2
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0answers
30 views
Does a map from a compact space to a filtered colimit factor through at a finite stage? [duplicate]
Let $K$ be a compact space, and let $A_i$ be a sequence of spaces in the following diagram.
$$A_0 \hookrightarrow A_1 \hookrightarrow A_2 \hookrightarrow \cdots$$
All the inclusions in the diagram are ...
3
votes
1answer
126 views
Reference request: Adjunction of $\Sigma^\infty$ and $\Omega^\infty$ is monoidal
I would like to see that the map of spectra $\Sigma^\infty \mathbb{C}P^\infty_+ \to KU$ is actually a map of ring spectra, where $KU$ denotes the complex K-theory spectrum and the map is given as the ...
2
votes
1answer
71 views
Splitting of a spectrum as a wedge
Suppose that $i:E\to F$ and $r:F\to E$ are maps of spectra ($S^1$-spectra of topological spaces) such that $r\circ i$ is a homotopy equivalence. Can we always show that the spectrum $F$ splits as a ...
1
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0answers
76 views
Stable Homotopy of Classical Groups
Let $\Gamma$ be the $2n \times 2n$ complex matrix
$$
\Gamma =
\begin{pmatrix}
U & V^* \\
V & U^*
\end{pmatrix}
$$
where $U$ and $V$ are $n \times n$ complex matrices. Now suppose $\Gamma$ ...
5
votes
1answer
179 views
Classifying map of tensor product of two line bundles
We know that $\mathbb{C}P^{\infty}$ is the classifying space of line bundles. Also we know that $\mathbb{C}P^{\infty}$ is an H space that is we have $$\mu: \mathbb{C}P^{\infty} \times \mathbb{C}P^{\...
3
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0answers
62 views
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 ]...
6
votes
2answers
116 views
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
0answers
65 views
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 ...
3
votes
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 ...
3
votes
0answers
82 views
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 ...
2
votes
0answers
49 views
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 ...
2
votes
2answers
141 views
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 ...
2
votes
1answer
126 views
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 ...
3
votes
0answers
95 views
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....
0
votes
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}} =(\...
2
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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,...
4
votes
0answers
78 views
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 $p-1$. ...
8
votes
0answers
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 ...
7
votes
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 ...
