# Questions tagged [stable-homotopy-theory]

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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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?
1 vote
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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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### 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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### 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 ...