Questions tagged [stable-homotopy-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
5 votes
0 answers
63 views

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
4 votes
1 answer
62 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
  • 10.6k
3 votes
1 answer
45 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
16 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
53 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
  • 3,304
1 vote
1 answer
82 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
  • 3,304
0 votes
1 answer
53 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
  • 3,304
1 vote
0 answers
35 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
  • 1,857
2 votes
0 answers
41 views

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
95 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
40 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
72 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
90 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
134 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
80 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
82 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
2 votes
1 answer
77 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
0 answers
92 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
53 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
  • 7,428
0 votes
1 answer
92 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
  • 7,428
0 votes
0 answers
54 views

homotopy inverse limit and smash product

Homotopy limits (in particular homotopy inverse limits) do not behave well with taking smash products. For a strongly dualizable spectrum $X$, and an inverse system of spectra$\{A_i\}_{i \in \mathbb{N}...
Algtop's user avatar
  • 25
1 vote
1 answer
77 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
2 votes
0 answers
96 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
87 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
136 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
  • 352
0 votes
0 answers
246 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
  • 352
2 votes
1 answer
232 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
  • 352
2 votes
1 answer
72 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
31 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
0 votes
1 answer
93 views

about loop and suspension

I am trying to understand the following construction. $X$ is a pointed CW complex. Define $Q(X) := hocolim_{n} \Omega^n \Sigma^n (X)$. Using the loop-suspension adjunction, we get maps like $X \to \...
Algtop's user avatar
  • 25
3 votes
1 answer
99 views

Defining the sheaf of bigraded homotopy groups in motivic homotopy theory.

I have been learning motivic homotopy theory from these notes and on page 154 (page 8 of the pdf), the author defines $\pi_{p,q}(E)$ where $E$ is an $(s,t)$-bispectrum. He defines it as the sheaf of ...
LoneStar's user avatar
  • 882
0 votes
0 answers
31 views

If $F$ is an $s$-spectrum, then what does the functor $\sum_s^n F$ mean for negative values of $n$?

I have been learning motivic homotopy theory from these notes and on page 151 (page 5 of the pdf), the author uses $\sum_s^n F$ where $F$ is an $s$-spectrum. For non-negative $n$, I figured that this ...
LoneStar's user avatar
  • 882
5 votes
1 answer
212 views

Stable homotopy equivalent but not homotopy equivalent

Are there known examples of spaces which are stable homotopy equivalent but not homotopy equivalent?
horned-sphere's user avatar
3 votes
1 answer
96 views

Stable Homotopy Inequivalence via Steenrod Squares

In Example 1.1 of https://maths.dur.ac.uk/users/andrew.lobb/master_morse.pdf it explains why $\mathbb{C}P^2$ and $S^2\vee S^4$ are not stably homotopy equivalent by looking at the second Steenrod ...
horned-sphere's user avatar
3 votes
1 answer
114 views

If $M$ is an $R$-module, how can I show that the Eilenberg-Maclane spectrum $HM$ is an $HR$-module spectrum

Let $R$ be a commutative ring with identity, and let $M$ be an $R$-module. I want to show that the Eilenberg-Maclane spectrum $HM$ is an $HR$-module spectrum. Specifically, I want to know how to ...
Austin Maison's user avatar
1 vote
0 answers
54 views

Are algebraic maps from the n-dimensional torus to the special unitary group of large enough degree null-homotopic?

Let $S^n\subset \mathbb{R}^{n+1}$ be the unit sphere, and $T^n=(S^1)^n$ the $n$-torus. Loday proved in 1 that every algebraic map $T^n\to S^n$ is null-homotopic. In particular, since $SU(2)\simeq S^3$,...
Vincent Nesme's user avatar
1 vote
1 answer
71 views

Proving the universal property for the localization functor $L_E$

I am trying to prove the following statement: If the functor $L_E$ exists, $(iii)$ for any map $g: X \to Y$ where $Y$ is $E_*$-local, there is a unique map $\tilde{g}: L_E X \to Y$ such that $\tilde{g}...
weird's user avatar
  • 29
0 votes
1 answer
105 views

Notation question: cohomology of a spectrum?

For curiosity's sake, I have been reading a bit about the history of the development of spectra, and in particular modern categories of spectra such as EKMM S-modules and diagram spectra (e.g., ...
crm114's user avatar
  • 23
0 votes
0 answers
37 views

Given spectrum and space $X$ construct chain complex that computes the cohomology

Suppose we are given a spectrum, as explained in https://en.wikipedia.org/wiki/Spectrum_(topology) this means we are given a collection of CW complexes $\{E_k\}_{k\in \mathbb N}$ and maps $i_k:\...
Overflowian's user avatar
  • 5,540
3 votes
1 answer
118 views

Proving some properties of the localization functor in the stable homotopy category.

I am trying to understand the paper named " Localization with respect to Certain Periodic Homology Theories" Here is the part of it I am trying to understand the proof of proposition 1.5 in ...
weird's user avatar
  • 29
2 votes
1 answer
697 views

How to learn motivic homotopy theory?

What prerequisite knowledge do I need to know to learn motivic homotopy theory? And what materials can I refer to to learn motivic homotopy theory?
Gustakovich's user avatar
2 votes
1 answer
101 views

Fibrations of spheres

I am reading Ravenel's "complex cobordism and stable homotopy groups of spheres". I am a bit confused by the notion of homotopy fiber, which basically gives a functorial way of regarding any ...
kindasorta's user avatar
  • 1,188
0 votes
0 answers
75 views

Does the generalized homology represented by the sphere spectrum give the stable homotopy groups on suspension spectra of pointed spaces?

If $\mathbb{S}$ is the sphere spectrum, I would like to show that the generalized homology theory represented by $\mathbb{S}$, when evaluated on the suspension spectrum $\Sigma^\infty X$ of a pointed (...
I.A.S. Tambe's user avatar
  • 2,401
1 vote
1 answer
108 views

Computing a Massey product.

Here is the question I am trying to solve: Can anyone help me in showing me how to compute this Massey Product?
Mathstupid's user avatar
3 votes
2 answers
210 views

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) ...
Qi Zhu's user avatar
  • 7,428
1 vote
0 answers
86 views

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 $(...
Jordan Levin's user avatar
0 votes
1 answer
119 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 ...
Keala's user avatar
  • 25
2 votes
1 answer
156 views

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^...
Keala's user avatar
  • 25
2 votes
0 answers
113 views

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?
Cille's user avatar
  • 401
2 votes
0 answers
195 views

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 ...
TTN's user avatar
  • 21