# Questions tagged [homotopy-theory]

Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.

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### How to learn homotopy theory

I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy ...
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### Rational G-spectrum and geometric fixed points

For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
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### Characterization of growth in terms of coarse algebraic topology

$$\newcommand{\mc}{\mathcal{#1}} \newcommand{\mbb}{\mathbb{#1}} \newcommand{\opn}{\operatorname{#1}} \DeclareMathOperator\cap{cap} \def\sse{\subseteq}$$ Coarse spaces Let $X$ be a coarse ...
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### Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?

Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic ...
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### Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?

A ring spectrum $E$ is complex oriented if it is equipped with a ring map $MU\rightarrow E$. It is complex orientable if such a ring map exists. An $E_\infty$-ring $E$ is $E_\infty$-complex oriented ...
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### When a model category with prescribed homotopy category exists?

My question is probably insanely hard or very well-studied, but I could not find an answer, so I will ask it here. Assume that we have a suitably complete closed module over $\operatorname{Ho}(sSet).$ ...
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### A topological tree is weakly contractible

Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
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### Inexistence of a Kan–Quillen model structure on globular sets

(This is in a sense a follow-up to my earlier question on a geometric definition of globular $\infty$-groupoids) We know by Scholie 8.4.14 of Cisinski's thesis that the globe category $\mathbb{G}$ is ...
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### Cyclic homology can be recovered from topological cyclic homology?

Let $R$ be a commutative ring, $S$ a $R$-algebra of finite type. By an equivalence of ring spectra $$\operatorname{HH}(S/R) \simeq \operatorname{THH}(S)\otimes_{\operatorname{THH}(R)} HR,$$ ...
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### If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory?

$\newcommand\Spt{\mathit{Spt}}\newcommand\GrAb{\mathit{GrAb}}$Let $A$ be a ring spectrum. Suppose that $A$ has a Künneth theorem — i.e. the homology theory $A_\ast : \Spt \to \GrAb$ is a strong ...
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### Decomposing a $\mathcal{M}$-valued presheaf into a homotopy colimit of representables

The context for this question comes from this arxiv preprint. Specifically, a remark in the final proof of the paper. To make the question more self-contained, I'll phrase this question in a slightly ...
1 vote
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### The key step in Serre's method on higher homotopy groups

Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups ...
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### Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum?

Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first ...
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