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Questions tagged [stable-homotopy-theory]

The 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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Why is the mapping spectrum $F(\Sigma^{\infty}_{+} X, E)$ a homotopy commutative ring spectrum?

Given a homotopy commutative ring spectrum $E$ and an anima (or space in $\infty$-categorical setting) $X$, the mapping spectrum $F(\Sigma^{\infty}_{+} X, E)$ is also a homotopy commutative ring ...
Si-Schao Lan's user avatar
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When is $\pi_{k}(X_{p}^{\wedge})$ not isomorphic to the p-completion of the group $\pi_{k}(X)$?

Let $X$ be a spectrum, $p$ a prime, we denote the $p$-completion of $X$ as $X_{p}^{\wedge}$, i.e. the Bousfield localization $L_{\mathbb{S}/p}X$. This $p$-completion of spectra is quite important in ...
Si-Schao Lan's user avatar
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106 views

Adjoint triplet induced by exact functor of stable categories

For $\mathscr{A}$ a small stable $\infty$-category, we can consider the following diagram: where $\mathcal{Y}_\mathscr{A}$ denotes the ordinary Yoneda embedding $\mathscr{A} \to \mathcal{P}(\mathscr{...
h3fr43nd's user avatar
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1 answer
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Failure of universal coefficient theorem for spectra

Let $X, E $ be spectra and assume that the smash product $X\otimes E$ is nullhomotopic. Does it follow that the mapping spectrum $\underline{map}_{Sp}(X,E)$ is also null-homotopic; is every map $X\to ...
Fabio Neugebauer's user avatar
1 vote
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Confusion of vanishing of motivic (co)homology

I'm trying to understand a vanishing result in Haesemeyer-Weibel as follows: Lemma $1.33$ For all smooth $X$ and $p > q$, we should have $Hom_{DM}(R, R_{tr}(X)(q)[p])=0$. From my understanding, the ...
user1346704's user avatar
4 votes
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Geometric fixed-points of MO

Recall that the value of the orthogonal spectrum $\mathbf{MO}$ at an inner product space $V$ is the Thom space of the tautological bundle over the Grassmannian of $|V|$-demensiomal planes in $V\oplus ...
yifan's user avatar
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the infinite category of pullback squares in an infinite stable category is also stable

I am currently reading Lurie's paper infinite stable category, in the proof of proposition 4.4, to show that every pushout square in an infinite stable category $C$ is also a pullback, he considers ...
Yang's user avatar
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Elementary proofs of the v1-periodicity of the Hopf map

I've been learning the periodicity theorem recently, and I know that the Hopf map $\eta \in \pi_1(\mathbb S)$ is $v_1$-periodic, which can be shown by machinery like ANSS. On the other hand, chromatic ...
Ziv's user avatar
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4 votes
2 answers
115 views

Stable and Pointed Infinity-Operads

I wanted to understand the construction of the maps $\mathsf{An}^{\times} \xrightarrow{(-)_+} \mathsf{An}_{*/}^{\wedge} \xrightarrow{\Sigma^{\infty}} \mathsf{Sp}^{\otimes}$ of commutative algebras in $...
Qi Zhu's user avatar
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3 votes
1 answer
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Homotopy in the space of linear isometries

I wanted to understand the linear isometries operad $\mathscr{L}$ as a model for $E_{\infty}$-operads for which I wanted to show that $\mathscr{L}(n) \simeq *$. This all reduces down to some ...
Qi Zhu's user avatar
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Point-set level object/category

What is the meaning behind the terminology "point-set level (object or category)” in context of stable homotopy theory? This appears e.g. in following excerpt quoted from Tom Bachmann's ...
user267839's user avatar
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1 vote
1 answer
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Generalized cohomology and stable homotopy groups of spectra

Suppose two spectra $E$ and $F$ have same stable homotopy groups $\pi_k$ for $k\geq0$, equivalently, $E^{-k}(S)=F^{-k}(S)$. If we replace the sphere spectrum $S$ by another spectrum $X$, I wonder if ...
Tovak's user avatar
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3 votes
1 answer
113 views

Based vs unbased spectra

I am trying to understand the potentially alternative definition of spectra in stable homotopy theory. I am assuming that everything is $\infty$-categorical. I will focus on finite spectra. One ...
user39598's user avatar
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1 vote
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Properties of colim Ωⁿ Σⁿ X

I am thinking about the paper of Gaunce Lewis Jr. showing the incompatibility of a certain five desirable properties of spectra. This paper makes me curious about the properties of $Q := \texttt{colim ...
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1 vote
1 answer
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connective spectra vs. spectra

I am thinking about infinite loop spaces and spectra. The category of connective spectra is in fact equivalent to the category of infinite loop spaces. Is it also the case that the category of spectra ...
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0 answers
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morphisms between stable unitary, orthogonal, and (compact) symplectic groups

The subgroup inclusions $$ U(N) \hookrightarrow Sp(N):=U(2N) \cap Sp(2N,\mathbb{C}), \quad U(N) \hookrightarrow O(2N) $$ induces some morphisms $$ f_1: U(\infty)\to Sp(\infty), f_2: U(\infty)\to O(\...
Hyeongmuk LIM's user avatar
2 votes
1 answer
83 views

Algebra with spectra: Vanishing in K(n) implies vanishing in a quotient of BP

I'm having a confusion about doing algebra with spectra. My question occured while reading p. 10 of Sanath Devalapurkar's Chromatic Homotopy Theory where we are in the course of proving the thick ...
Qi Zhu's user avatar
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1 vote
0 answers
49 views

Spectral Sequences from Two Kinds of Filtrations

Reading up on spectral sequences I found out that many spectral sequences can be obtained in two different ways, either by a filtration on the object or by a filtration on the invariant to be ...
Jonas Linssen's user avatar
7 votes
1 answer
142 views

A possible error in May's Concise Algebraic Topology (prespectra)

In chapter 22.1 of May's A Concise Course in Algebraic Topology, he claims that the prespectrum $\{T_n\}$ of spaces where each $T_n$ is $(n-1)$-connected yields a reduced homology theory by setting $\...
Emory Sun's user avatar
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6 votes
1 answer
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A definition for sheaves on stacks

Here's is a definition of sheaves on stacks in the famous Complex Oriented Cohomology Theories and the Language of Stacks (COCTALOS) lecture notes by Mike Hopkins. Since it was written by students and ...
Qi Zhu's user avatar
  • 8,358
1 vote
2 answers
114 views

Ranking topological invariants by "strength?"

Is it possible to rank the common topological invariants (homology, cohomology, homotopy) according to their "strength?" By this I mean, can spaces have different homotopy groups yet the ...
raynea's user avatar
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Being a triangulated functor is a property or an additional structure?

Let $\mathcal T$ and $\mathcal T'$ be triangulated categories and consider an additive functor $F:\mathcal T \to \mathcal T'$. Some authors say $F$ is triangulated if there exists a natural ...
P. Usada's user avatar
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Mod p homology of extended powers

In his 1973 paper "The nilpotency of elements of the stable homotopygroups of spheres", Goro Nishida calculates, among other things, the homology of extended powers. Letting $M_k = S^k \...
Julius J.'s user avatar
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4 votes
1 answer
91 views

Does shift (suspension) commute with mapping cone in homological algebra?

I'll be using the conventions from Kashiwara & Schapira's Categories and Sheaves book here. Let $\mathsf{A}$ be an additive category with translation $S: \mathsf{A} \xrightarrow{\sim} \mathsf{A}$. ...
ಠ_ಠ's user avatar
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3 votes
1 answer
67 views

Rational homotopy groups of spectra

It is claimed in a paper of Adams, Harris and Switzer that $$\pi_*E \otimes \pi_*F \otimes \mathbb{Q} \to E_*F \otimes \mathbb{Q}$$ is an isomorphism. This map is constructed by taking the map $\pi_*E ...
categorically_stupid's user avatar
0 votes
0 answers
18 views

The Relationship Between Two Constructions of Topological Modular Forms

There are two explicit constructions of topological modular forms. One is in the 12th section of the book $Topological$ $Modular$ $Forms$, and the other is Lurie's Elliptic Cohomology II in his ...
user884626's user avatar
1 vote
1 answer
60 views

Canonical morphism $F(X,I)\otimes Y\to F(X,Y)$ in a closed symmetric monoidal category

Let $(\mathcal{C},\otimes,I,F)$ be a closed symmetric monoidal category, where $F$ is the 'internal Hom' functor, $I$ is the unit object and $\otimes$ is the monoidal product. I am reading Definition ...
user829347's user avatar
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1 vote
1 answer
228 views

What is the '$p$-completion of a spectrum'?

Suppose I have a finite pointed CW complex X and an integer $n$. In the context of stable homotopy theory this is known as a finite spectrum. For a given prime $p$, something called the $p$-completion ...
user829347's user avatar
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0 votes
1 answer
77 views

Using Freudenthal Suspension Theorem to construct the Spanier-Whitehead category

I am trying to understanding the construction of the Spanier-Whitehead category, and the role of the Freudenthal Suspension Theorem in this. FST seems to take many forms, but here is the one that I ...
user829347's user avatar
  • 3,440
1 vote
0 answers
50 views

Transfer map for the Mackey functor $\underline{\pi}_n^H$.

Let $G$ be a finite group ad let $X$ be a $G$-space. Consider the following Mackey functor, that I will denote by $\underline{\pi}_n$: $G/H\mapsto \pi_n^H(X)$, where $\pi_n^H(X)$ refers to the stable ...
Dog_69's user avatar
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2 votes
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p-completion preserves cofiber sequences

Suppose I have a cofiber sequence $X \to Y \to Z$ of spectra in the stable homotopy category. I want to show that there is still a cofiber sequence $X^\wedge_p \to Y^\wedge_p \to Z^\wedge_p$ after p-...
categorically_stupid's user avatar
3 votes
0 answers
107 views

Proof of characterization of $E_1$ page of the Adams spectral sequence

I am following the nLab notes on the Adams spectral sequence, as they seem to be the most detailed I can find. That being said, I am still struggling to understand many of the steps. I have explained ...
Isaiah Dailey's user avatar
1 vote
1 answer
47 views

Can categories with suspension be considered as Cat-enriched presheaves over certain strict 2-category?

I came across the concept of a category with suspension $(\mathcal{C},\Sigma)$ which is defined as a category $\mathcal{C}$ together with an endofunctor $\Sigma:\mathcal{C}\rightarrow \mathcal{C}$. We ...
Zhenhui Ding's user avatar
4 votes
0 answers
77 views

The stable category of $\mathbb{Z}$

Is there an alternative description/characterization of the stable module category of Abelian groups? I guess that the category of torsion groups is a subcategory of it, but is it all of it? What is ...
Michal's user avatar
  • 51
4 votes
1 answer
109 views

Showing that the comultiplication map $E_*E\to E_*E\otimes E_*E$ is co-associative for a flat homotopy commutative ring spectrum

Let $(E,\mu,e)$ be a flat homotopy commutative ring spectrum, so we have an isomorphism $$\Phi_E:E_*E\otimes_{\pi_*E}E_*E\to E_*(E\wedge E)$$ sending homogeneous elements $x:S^n\to E\wedge E$ and $y:S^...
Isaiah Dailey's user avatar
2 votes
3 answers
263 views

Homotopy groups of wedge sums of spectra

This question came up when I was trying to understand Lemma 2.2.9 in Barnes & Rotzheim, which states that for any set of (sequential) spectra $X_i$, the natural map $$\bigoplus_i\pi_n(X_i)\to\pi_n\...
Tipping Octopus's user avatar
1 vote
1 answer
119 views

Is the full subcategory of $p$-local finite spectra a thick subcategory of all finite spectra?

I am trying to understand Balmer's classification of the spectrum of the category $\mathsf{Sp}^\text{fin}$ of finite spectra. The inclusion $\mathsf{Sp}^\text{fin}_{(p)} \subseteq \mathsf{Sp}^\text{...
Jonas Linssen's user avatar
2 votes
1 answer
146 views

Why does a stable category admitting finite limits, filtered colimits and $\Sigma$ admit pushouts?

I'm trying to understand how to compute pushout in Spectra. The reason it should satisfy it is because a stable category admitting finite limits, filtered colimits and $\Sigma$ (i.e pushout of $X \to *...
user135743's user avatar
3 votes
1 answer
138 views

When is a $p$-local spectrum zero?

I am currently reading Lurie's notes on chromatic homotopy theory and fail to see the following remark in lecture 26: Remark 5. Let $X$ be a finite $p$-local spectrum. Then $H_\ast(X,\Bbb F_p) \simeq ...
Jonas Linssen's user avatar
5 votes
1 answer
107 views

Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces

Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
Mathematics enthusiast's user avatar
0 votes
1 answer
59 views

Induction preserves weak equivalences

Let $G$ be a finite group and $H \leq G$ be a subgroup. There is an induction functor $G \ltimes_H - : \mathbf{Sp}^H \to \mathbf{Sp}^G$ from the category of $H$-spectra to the category of $G$-spectra (...
Qi Zhu's user avatar
  • 8,358
0 votes
1 answer
129 views

Geometric Fixed Points of Thom Spectrum

Recall that the Thom spectrum is an orthogonal spectrum $\operatorname{mO} \in \mathbf{Sp}$ defined via $\operatorname{mO}(V) = \operatorname{Th}(\operatorname{Gr}_{\dim{V}}(V \oplus \mathbb{R}^{\...
Qi Zhu's user avatar
  • 8,358
1 vote
1 answer
134 views

Stable homotopy groups commute with inverse limit

Suppose we have a family of spectra $(E_i)_{i \in I}$ such that the inverse limit $\lim_i E_i$ does exist in the stable homotopy category (i.e. $\lim_i E_i$ is the limit in $\mathrm{SHC}$, the stable ...
Candyblock's user avatar
3 votes
0 answers
157 views

Ring structure on the stable homotopy groups of spheres well-defined?

The stable homotopy groups of the spheres $\pi_{*}^{s}$ assemble into a graded ring $\pi_{*}^{s} = \bigoplus_{n\geq 0} \pi_{n}^{s},$ with the graded product defined `in terms of composition'. For ...
Sunny Sood's user avatar
4 votes
0 answers
100 views

Cotensor product of Hopf algebroids constructed out of Brown-Peterson Spectrum

I am reading Ravenel's green book(Complex Cobordism and Stable Homotopy Groups of Spheres), there is an example in its 306 page: Let $(A, \Gamma) := (\pi_* BP, BP_* BP) \cong (\mathbb{Z}_{(p)}[v_1, ...
Cloudifold's user avatar
0 votes
1 answer
140 views

Obtaining the homotopy groups of a spectrum from the Adams spectral sequence

I am following the two examples in Masulli's document, and even though everything is very clear I can't figure out a small detail. In page 25 he goes: It can be shown that the vertical lines in the ...
groupoid's user avatar
  • 372
0 votes
0 answers
274 views

Computing the stable homotopy groups of the spheres using the Adams spectral sequence

As an example of application of the Adams spectral sequence I've encountered the computation of the stable homotopy groups of the sphere. This spectral sequence says that $\textrm{Ext}^{s,t}_\mathcal{...
groupoid's user avatar
  • 372
2 votes
1 answer
268 views

Why is the Steenrod algebra isomorphic to the cohomology of the Eilenberg MacLane spaces?

I will stick to $p=2$. I define the Steenrod algebra to be the algebra of (topological) stable cohomology operations modulo 2. I've found in the literature the identification of the Steenrod algebra $\...
groupoid's user avatar
  • 372
2 votes
1 answer
75 views

Stable homotopy type of a space

Is it possible to get a space (may not be a CW complex) which has some non zero homotopy group, but all of whose stable homotopy groups are zero?
Algtop's user avatar
  • 25
1 vote
0 answers
36 views

conditions for maps between homotopy colimits

Given two diagrams $F_1$ and $F_2$, $C \to Top_{*}$, is there any sufficient condition for the existence of a continuous function on $hocolim(F_1) \to hocolim(F_2)$ where $C$ is a small category. Of ...
Monkey.D.Luffy's user avatar

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