Questions tagged [stable-homotopy-theory]
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192
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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 \...
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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 $S: \mathsf{A} \xrightarrow{\sim} \mathsf{A}$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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^...
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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 $$\bigoplus_i\pi_n(X_i)\to\pi_n\...
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Is the full subcategory of $p$-local finite spectra a thick subcategory of all finite spectra?
I am trying to understand Balmer's classification of the spectrum of the category $\mathsf{Sp}^\text{fin}$ of finite spectra.
The inclusion $\mathsf{Sp}^\text{fin}_{(p)} \subseteq \mathsf{Sp}^\text{...
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Why does a stable category admitting finite limits, filtered colimits and $\Sigma$ admit pushouts?
I'm trying to understand how to compute pushout in Spectra.
The reason it should satisfy it is because a stable category admitting finite limits, filtered colimits and $\Sigma$ (i.e pushout of $X \to *...
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When is a $p$-local spectrum zero?
I am currently reading Lurie's notes on chromatic homotopy theory and fail to see the following remark in lecture 26:
Remark 5. Let $X$ be a finite $p$-local spectrum. Then $H_\ast(X,\Bbb F_p) \simeq ...
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Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
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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 (...
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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}^{\...
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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}...
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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 ...
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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 ...
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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, ...
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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 ...
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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{...
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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 $\...
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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?
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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 ...
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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 \...
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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 ...
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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 ...
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Stable homotopy equivalent but not homotopy equivalent
Are there known examples of spaces which are stable homotopy equivalent but not homotopy equivalent?
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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 ...
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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 ...
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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 $SU(2)\simeq S^3$,...
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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}...
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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., ...
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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:\...
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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 ...
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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?
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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 ...
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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 (...
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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?
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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) ...
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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 $(...
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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 ...
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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^...
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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?
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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 ...
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