# 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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### Infinite loop space as an endofunctor of compactly generated weak hausdorff topological spaces?

I am trying to see whether it is possible to define smash product of infinite loop spaces using the space $S^{\infty}$. Let C be the category of compactly generated weak Hausdorff topological spaces. ...
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### Reference request for equivalences between different models of Lax limits

There are several models for Lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
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### Can a phantom map have finite cofiber?

Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum? Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
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### Is the mapping cylinder a replacement for morphism by cofibration in model categories?

Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
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### The James and Morse filtrations of homotopy groups

Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
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### On the initiality of the inclusion from the simplex category to the paracycle category

Thm B.3 of Nikolaus and Scholze shows that the natural inclusion $\Delta \to \Lambda_\infty$, from the simplex category to the paracycle category, is an initial functor, i.e. satisfies the hypotheses ...
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### Does $\pi_1(H)=0\Rightarrow \pi_3(G/H)=0$ for a simple and simply connected Lie group $G$?

$\DeclareMathOperator\SU{SU}$Let $G$ be a simple and simply-connected Lie group and $H\neq 1$ be a simple and simply connected subgroup, is it true that $\pi_3(G/H)=0$? If not, what is a counter-...
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### Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres

There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...
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### Two $E_\infty$ structures on infinite matrices

Let $O$ be the infinite orthogonal group. By taking a colimit of the diagram of topological groups $O(1) \to O(2) \to O(3) \to \ldots$, we know $O$ has a continuous group operation given by matrix ...
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### Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$

I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3: Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
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### Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?

It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
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### A question on the manifold $\{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\}$

Consider a manifold $N$ defined as follows $$N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3},$$ where $S^2$ denotes the two dimensional sphere, $(\cdot,\cdot)$ ...
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### What are some toy models for the stable homotopy groups of spheres?

The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero. Question: What are some "toy models" ...
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### What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category). Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...
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### Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
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### What is the Goldie dimension of the ring of stable stems?

Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
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### The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
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