Questions tagged [stable-homotopy-theory]

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3
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0answers
31 views

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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45 views

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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30 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 ($1.5$), ...
2
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0answers
48 views

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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40 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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1answer
36 views

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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1answer
39 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)...
4
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54 views

$KO_*$ 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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15 views

Cofibre of ring spectra

I was reading the following paper about how to generalise the classical derived algebra to the setting of spectra https://arxiv.org/pdf/1601.02473.pdf and I cannot understand a passage not explained ...
2
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1answer
33 views

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 ...
3
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0answers
41 views

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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20 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 ...
1
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1answer
24 views

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 ...
2
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1answer
59 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....
2
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1answer
46 views

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 ...
3
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0answers
48 views

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 ...
2
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1answer
40 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(...
2
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1answer
50 views

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 ...
2
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61 views

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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34 views

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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1answer
184 views

Smash product of $S^1$ with the interval $I$

I'm trying to work through various examples of smash products of spaces. In order to check to see if what I'm doing is correct, is the smash product $S^1 \wedge I$ of the circle with the interval $[-1,...
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0answers
23 views

Values of the $\mathbb{Z}/2$-spectrum $KR$ of $K$-theory with Reality

Atiyah's $K$-theory with Reality produces a $\mathbb{Z}/2$-spectrum $KR$. But I am stuck as I don't really know what the values of this equivariant spectrum should be on representations. Any ...
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25 views

The $K(n)$-local category has no non-trivial localizing subcategories

Let $Sp$ be the stable homotopy category and $K(n)$ the $n$-th Morava $K$-theory. Then we set $L_{K(n)}Sp$ to be the $K(n)$-local category. It is folklore that this has only trivial localizing ...
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0answers
65 views

Does an $X$ exist so $X \simeq \Omega^n Y$ for all $n$, and X is not an infinite loop space?

Are there examples of spaces $X$ that have the property $X \simeq \Omega ^n Y$ with $Y$ depending on $n$, for all natural $n$, with $X$ not an infinite loop space? I have a feeling these things ...
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0answers
49 views

Commutativity of monoid object on $\mathcal{D}$-spaces

Let ($\mathcal{D}, \otimes, 0)$ be a symmetric monoidal category enriched over pointed spaces, and denote by $\mathsf{Top}_*^\mathcal{D}$ to the category of "$\mathcal{D}$-spaces", the category of ...
3
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0answers
60 views

Filtered colimits in Adams' category

I am currently reading Part 3 of Adams's book "stable homotopy and generalized cohomology", and I got stuck when following his argument. In Proposition 5.4, he states that when $W$ and $X$ are finite ...
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1answer
32 views

Reference for the cohomology of SU

Let SU be the infinite special group. Where can I find the following fact (state in part III of the Adam's blue book): $H^{6}(SU,Z)=0$. Thank you.
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1answer
44 views

Factorization of the orientation map $MU\to H\mathbb{Z}$ through $ku$?

Let $MU$ denote the complex cobordism spectrum and $ku$ the connective cover of the complex $K$-theory spectrum. Is it true that the orientation map $MU\to H\mathbb{Z}$ factors through $ku$?
3
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1answer
76 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
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1answer
394 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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58 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
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1answer
79 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
64 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
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1answer
90 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
196 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-...
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0answers
151 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
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1answer
325 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
66 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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0answers
21 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
248 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
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1answer
104 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
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0answers
139 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 $\...
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1answer
71 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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53 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. ...
5
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1answer
123 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), \...
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1answer
52 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 @>&...
3
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3answers
131 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
61 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 $[A \wedge M, B]\cong [A, \Sigma^{-1}M \wedge B]$,...
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0answers
105 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). $...
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0answers
41 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, ...