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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0answers
67 views

2d TQFTs with values in simplicial sets and Reedy categories

Let $Cob$ be the category such that $Obj(Cob)$ is $\emptyset\sqcup\mathbb{N}$, with $n$ seen as the union of $(n+1)$ circles numbered from $0$ to $n$, morphisms are (homeomorphism classes of ...
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What is the inverse in K-theory represented by Clifford module extensions?

I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X_n]$ where the ...
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463 views

Étale homotopy type of $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_\infty \}$

Has anyone formally calculated the étale homotopy type of $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\infty} \}$? According to arithmetic topology, $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\...
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1answer
138 views

Filling condition for quasi-categories

I am trying to get a better understanding of how the inner horn filling condition in higher cells corresponds to higher associativity laws. For instance, I am trying to understand how the difference ...
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5answers
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What is the cotangent complex good for?

The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
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Products of cones and cones of joins

The join of $A$ and $B$ is the pushout of the diagram $$ CA \times B \gets A\times B \to A\times CB, $$ which can be formulated in either the pointed or unpointed topological category. This pushout is ...
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297 views

Homotopical characterization of CW complexes

Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$. Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
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169 views

Homotopical characterization of manifolds

Let $X$ be a compact metrizable topological space of covering dimension $4$. Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...
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Fundamental group and countability

A friend of mine on a Discord server talked about an exercise she had to do (she's in master's degree): prove that you can put an uncountable number of disjoint "5" in the euclidean plane $\...
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Recovering operad units from homotopy units

It is my understanding that the $\infty$-category of non-unital connected topological monoids is equivalent to the $\infty$-category of connected topological groups. It follows that the functor from ...
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Complement of contractible locally Euclidean subspace

Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
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137 views

Homotopy groups of ball complement

Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$. Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
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Every Spectral Deligne-Mumford stack satsifies fpqc descent?

In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too ...
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1answer
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Different ways to “deloop” a (topological) $A_\infty$-algebra

Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$: Rectify $X$ by taking the ...
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131 views

Covers of a 4-manifold pull back a cohomology class to any algebraic multiple

Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$? Is ...
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1answer
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Smooth covers pulling back a cohomology class to any algebraic multiple

Fix an algebraic integer $x\neq 0$. Does there exist a closed smooth manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
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1answer
305 views

Smooth complex projective surface as the total space of a Serre fibration

Let $M$ be the underlying topological manifold of a smooth complex projective surface. Assume $\pi_1(M)=\{0\}$ and $\pi_2(M)\neq \mathbb{Z}^2$. Is there a Serre fibration $M\to B$ where $B$ is a CW ...
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261 views

It is possible that $ X \simeq ΩX $? and that $ X \simeq Ω^ 2X $?

Original post: https://math.stackexchange.com/questions/3810423/it-is-possible-that-x-simeq-%ce%a9x-and-that-x-simeq-%ce%a9-2x I am studying J. Strom's Modern Classical Homotopy Theory. In chapter 4 ...
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1answer
119 views

$(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an equivalence

Let $\mathbb{Z}/p^{\infty}$ denote the Prufer group. By $p$-completion properties, it follows that $(B\mathbb Z/p^{\infty})^{\wedge}_p\simeq K(\mathbb{Z}^{\wedge}_p,2)\simeq(BS^1)^{\wedge}_p$. But, ...
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Classifying spaces of amalgamated topological monoids

Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
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Borel conjecture and arbitrary surface

Before starting my question I want to write something that I already know. Borel Conjecture: Any homotopy equivalence between two closed aspherical manifolds is homotopic to a homeomorphism. Now, my ...
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1answer
336 views

Milnor excision for algebraic stacks

Recall that a commutative square of commutative rings $$\begin{matrix} A&\to&B\\ \downarrow &&\downarrow\\ A^\prime&\to&B^\prime\end{matrix}$$ is called a Milnor square if the ...
32
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1answer
347 views

Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$

Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $A$, Eilenberg and Mac Lane define its cubical construction $QA$ to be a certain connective chain complex, ...
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2answers
273 views

Contractibility of the category of cosimplicial resolutions

Let $\gamma : \mathcal{C} \to \mathcal{M}$ be a functor and define a cosimplicial resoultion of $\gamma$ as a functor $\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$ such that $\Gamma C$ is Reedy ...
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111 views

Is an (n-1)-sphere quotient by an (n-1)-sphere contractible? [closed]

I am thinking about the homotopy type of the following quotient space: Let $X$ be a topological space and $A$ be a subspace of $X$. If both $X$ and $A$ have homotopy type of a sphere $S^{n-1}$ (of the ...
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308 views

Does every map $K(\mathbb{Z}, n) \to K(\mathbb{Z}/m, n + k)$ factor through $K(\mathbb{Z}/m, n)$?

Recall that a cohomology operation is a natural transformation $H^n(-; \pi) \to H^{n+k}(-; G)$ defined on CW complexes. Does every cohomology operation $H^n(-; \mathbb{Z}) \to H^{n+k}(-; \mathbb{Z}/m)...
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115 views

A question on recognition of equivariant loop spaces

I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places. We know from the work of Segal that to give a loop ...
3
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162 views

Étale homotopy equivalent varieties are deformation equivalent

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $V_1$ and $V_2$ be étale simply-connected smooth proper varieties over $k$. Assume there is an isomorphism between the prime-to-...
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2answers
361 views

Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?

An exceptional complex Lie algebra is a simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five such Lie algebras: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}...
7
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1answer
288 views

When does a triangulated category have a heart?

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that ...
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221 views

Is there an exponential map in $\mathbb A^1$ homotopy theory?

Let $k$ be a field, and let $Z \subset X$ be a smooth subscheme of a smooth scheme $X$. When $k = \mathbb C$, there is a distinguished isotopy class of (topological) open embeddings $N_Z \to X$. In ...
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2answers
439 views

On the link between homology and homotopy

In the last semester I learned homological algebra and higher category theory/homotopy theory. But I am kind of confused when I try to really understand the link between the two subjects (this is ...
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3answers
289 views

Embedded ribbons and regular isotopy

I'm reading Kauffman's 1990 paper "An Invariant of Regular Isotopy" about knots that are isotopic through only Reidemeister Type II and III moves, which is known as a regular isotopy. His ...
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171 views

Hopf invariants of elements from spherical fibrations

Let $G_n$ be the space of homotopy-equivalences of $S^{n-1}$. Evaluation produces a map $G_{n} \to S^{n-1}$. For $n = 2m+1$, I would like to understand the induced map on $\pi_{4m-1}$. More precisely, ...
6
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1answer
308 views

Any continuous map is homotopic to one assuming fixed values at finitely many points

Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected. Given $n$ pairs of points $(x_i, y_i)$ where $x_i\in X$ and $y_i\...
4
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64 views

Free abelian group on a space and fibrations

Let $X$ be a topological space. Endow the free abelian group on $X$, $\mathbb Z[X]$, the quotient topology coming from the surjection $\bigsqcup_n X^n \times \mathbb Z^n \to \mathbb Z[X]$. For $Y$ a ...
4
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0answers
124 views

Interlocking (weak) factorization systems

I'm interested in instances of the following data: $C$ is a (possibly higher) category; $(L,M)$ is a weak factorization system (wfs) on $C$; $(M,R)$ is a unique factorization system (fs) on $C$. ...
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159 views

When every closed and connected subset is path connected

Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
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91 views

About Homotopy Transfer Lemma

If M, A are two differential graded complexes over a commutative ring R with the following data, $$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ ...
4
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1answer
187 views

Homotopy descent and cohomology

I've been reading "Structured Brown representability via concordance" by D.Pavlov (https://dmitripavlov.org/concordance.pdf) and I'm strugeling with a point and was wondering if someone ...
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200 views

One periodic cohomology theories?

Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. ...
6
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1answer
159 views

Module spectrum maps up to stable homotopy

Let $R$ be a commutative ring spectrum, $M$ and $N$ be a $R$-module spectra. Let us consider $R$-module maps from $M$ to $N$ up to stable homotopy, that is maps $M \to N$ such that the composites $R \...
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138 views

Intereresting classes of topological spaces locally modelled on some fixed spaces

A substantial part of mathematics studies manifolds which are defined as second countable Hausdorff locally Euclidean topological spaces. That always seemed kind of random to me since what is so ...
9
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1answer
253 views

Intuition for categorical fibrations?

I think I have a pretty good intuitive understanding of most types of fibrations of quasicategories: a (trivial) Kan fibration is a bundle of (contractible) spaces with equivalent fibers, a left/...
6
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1answer
218 views

(Pro-)representable functors and full subcategories in homotopy theory

$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{...
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2answers
818 views

If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic

I am currently reading Kervaire-Milnor's paper "Groups of Homotopy Spheres I", Annals of Mathematics, and I am trying to prove (or disprove) the following result. The more elementary the ...
8
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0answers
94 views

Every immersion can be deformed to have only transverse self-intersections

I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it. Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true ...
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0answers
212 views

Étale homotopy type of (derived) loop space

A feature of derived algebraic geometry is that we have internal homs. Furthermore, we can think of $B\mathbb{Z}$ as the derived algebraic geometric analogue of $S^1$. Thus we have an analogue of the ...
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0answers
93 views

Essential Image of the Étale Homotopy type

For any scheme $X$ we can associate the étale homotopy type $Et(X)$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential ...
18
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1answer
479 views

Homotopy equivalent Postnikov sections but not homotopy equivalent

Two pointed, connected CW complexes with the same homotopy groups need not be homotopy equivalent (Are there two non-homotopy equivalent spaces with equal homotopy groups?). Moreover, having the same ...

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