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Questions tagged [homotopy-theory]

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called a homotopy group can be obtained from the equivalence classes. The simplest homotopy group is the fundamental group. Homotopy groups are important invariants in algebraic topology.

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Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
Zhen Lin's user avatar
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Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
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Relation between non-vanishing Vector Fields on $\mathbb{T}^2$ and Fundamental Group Maps

Let X be a vector field on $\mathbb {T}^2$, we say that $\varphi: \mathbb {R} \to \mathbb {T}^2$ is a periodic orbit of $X $, if $\varphi $ is a periodic function and $\varphi'(t) = X (\varphi(t)), \...
Matheus Manzatto's user avatar
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Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
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Do Hopf bundles give all relations between these "composition factors"?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but ...
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How to prove that the center of the fundamental group of $T_g$ is trivial for $g \geq 2$?

Where $T_g$ is a closed orientable surface of genus g. I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization ...
Infinity's user avatar
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Whitehead product and a homotopy group of a wedge sum

Note : this question has been crossposted on the mathematics Overflow. Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goal is to prove ...
Anthony's user avatar
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13 votes
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833 views

Visualising $\pi_2(S^2)$ and $\pi_2(\mathbb{R}P^2)$

I would like to know how to visualise the elements of $\pi_2(S^2)$ as unit vector fields on the sphere. For instance, the generator $a$ of $\pi_2(S^2)$ would be visualised as a 'hedgehog' ...
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Are neighbourhood deformation retracts transitive?

I have the following definition of a neighbourhood deformation retract (or NDR, for short): A pair $(X, A)$ is called an NDR if $A\subseteq X$ is closed and there exists a neighbourhood $A\subseteq V\...
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What is the essential difference between a sheaf and a fibration with lifting property?

Are there cases where the two notions coincide? By sheaf I mean a pre-sheaf ($\mathcal{C}^{op} \rightarrow \mathcal{Set}$) satisfying the sheaf condition. The sheaf condition says that, for every ...
Yan King Yin's user avatar
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Many definitions of Hochschild homology and cyclic homology

It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some ...
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Space with semi-locally simply connected open subsets

A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
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Higher homotopies

It is a standard fact that two morphisms $f,g \in S_1$ (with the same vertexes) in an $\infty$-category $S$ are homotopic in the sense that there is a $2$-simplex $\sigma \in S_2$ such that $d_0\...
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Difference between $[pt/G]$ and $BG$

Let $G$ be a finite group. In topological category, we have the quotient stack of a point by $G$, denoted by $[pt/G]$. We also have the classifying space $BG$, which is a topological space. I am a ...
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Map $\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y))$ bijection?

I have two questions. How do I see that the map$$\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y)), \quad [f] \mapsto f_*,$$is a bijection if $X$ is an $(n - 1)$-connected CW complex and $Y$ is ...
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Homotopy colimits and homotopy fibers

Let $\dots\to X_n\to X_{n+1}\to \dots$ and $\dots\to Y_n\to Y_{n+1}\to \dots$ be a sequences of (fibrant) simplicial sets and $f_n:X_n\to Y_n$ are fibrations that commute with maps in sequences. Let $...
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What were Poincaré's fundamental groups? (a motivation request)

In this math.overflow question: https://mathoverflow.net/questions/143116/where-does-the-notation-pi-1x-x-for-the-fundamental-group-first-appear, an answer posits that Poincaré regarded the ...
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chain homotopy equivalence and quasi-isomorphism

Suppose $(C,d)$ and $(D,\delta)$ are two chain complexes over a field and $f:C\to D$ is a chain map. We say $f$ is a quasi-isomorphism if it induces an isomorphism of the homology groups $H(C,d)\to ...
Hang's user avatar
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Where to learn homotopy theory?

I've discovered recently that homotopy is more powerful than I thought. I have some knowledge about classic homotopy theory on topological spaces and simplicial complexes and very little about ...
Alberto Macías's user avatar
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343 views

Does a group homomorphism up to homotopy induce a map between classifying spaces?

Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces. If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces $$Bf\...
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Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
Julian Kniephoff's user avatar
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408 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
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Why is well-pointedness necessary for $X\hookrightarrow M_f$ to be a pointed cofibration?

In Jeffrey Strom's Modern Classical Homotopy Theory on page $125$, it is stated that "Now we come to one of the crucial differences between the pointed and the unpointed categories. The mapping ...
Olivier Bégassat's user avatar
8 votes
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252 views

Visualizing the Poincare homology sphere

I know that past a certain point, one should graduate from the view that homology/homotopy groups "count holes" in any realistic, grounded, real-life meaning of the word "hole". ...
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8 votes
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111 views

Is every homotopy equivalence a fibration over the interval?

It is a well known fact from homotopy theory that if $X\to Y$ is a Hurewicz fibration and $y_0,y_1$ are two points of $Y$ connected by a path $y_0\sim y_1$, then the fibers over $y_0$ and $y_1$ are ...
Robert Szafarczyk's user avatar
8 votes
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253 views

Homology and the reduced A-linearization of a space.

On page 16 of his book on symmetric spectra, Stefan Schwede defines the $n$-th Eilenberg-Mac Lane space $(HA)_n$ of an abelian group $A$ by means of a construction called the reduced $A$-...
merle's user avatar
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Is the nerve of a symmetric monoidal category a K-theory space?

It's an amazing fact that if the $C$ is a symmetric monoidal category so that its components form a group, then $NC$ is an infinite loop space. Now if we have a Waldhausen category $D$, a category ...
Connor Malin's user avatar
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8 votes
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340 views

Understanding suspension in the model category of chain complexes.

In a model category $C$ and $X \in C$ we can define $\Sigma X$ as the homotopy pushout of $$* \leftarrow X \rightarrow * $$ letting $*$ denote the terminal object. The way I understand it is that if $...
Noel Lundström's user avatar
8 votes
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141 views

Intuition for Freudenthal Suspension

One version of the Freudenthal suspension theorem is the following: Suppose a CW complex $X$ is a union of two subcomplexes $A,B$ with $A\cap B\neq\emptyset$ connected and nonempty. If $(A,A\cap B)$ ...
pancini's user avatar
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Replacing the fibre of a fibration.

Let $p:E\rightarrow X$ be a fibration over a pointed, connected CW complex $X$ with typical fibre $F=p^{-1}(\ast)$. Given another space $F'$ and a homotopy equivalence $F\simeq F'$, is it possible to ...
Tyrone's user avatar
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106 views

CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
Tyrone's user avatar
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8 votes
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Homotopy of homeomorphisms implies homeomorphism $X \times [0,1] \to X \times [0,1]$?

Let $h_0$ and $h_1$ be self-homeomorphisms of a topological space $X$. Let us say that $h_0$ is homotopic to $h_1$, and write $h_0 \sim h_1$ if there exists a 1-parameter family $h_t$, $t \in [0,1]$ ...
Mike F's user avatar
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8 votes
0 answers
420 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
Tom's user avatar
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8 votes
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228 views

Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow K_3$,...
Samarkand's user avatar
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automorphisms of the torus bundle over the circle

Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix. A fiber-preserving homeomorphism of $\xi$ is called automorphism of ...
Gleb's user avatar
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8 votes
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185 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
Gina's user avatar
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7 votes
0 answers
328 views

Five lemma for homotopy exact sequences of triple

Suppose we have topological spaces $B\subset A\subset X$ and $B'\subset A'\subset X'$, and a continuous map $f:X\to X'$ with $f(A)\subset A'$, $f(B)\subset B'$. Consider the homotopy long exact ...
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7 votes
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176 views

Homotopy of integral curves of a gradient field preserves levelsets (?)

Given a differentiable scalar field $f$ on a Riemannian manifold $X$ (with properties as required) I would like to formulate a homotopy of maximal integral curves $\gamma, \tilde{\gamma}$ of the ...
stewori's user avatar
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0 answers
154 views

$Mat_{ \infty,n }(\mathbb C)$ of max rank is contractible

We want to show that for a given $n\in\mathbb{N}$, the set of all the $\infty\times n$ matrices $Mat_{\infty,n}(\mathbb{C})$ of max rank $n$ form a contractible space. This set is obtained by taking ...
Maffred's user avatar
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7 votes
0 answers
275 views

Example of a Serre fibration between manifolds which is not a fiber bundle?

I'm looking for an example of a map $f : X \to Y$, where $X$ and $Y$ are manifolds (without boundary), and $f$ is a Serre fibration, but $f$ is not a fiber bundle. I know that if $f$ is proper, and $...
Elle Najt's user avatar
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7 votes
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511 views

Fundamental Group of a Quotient of an Annulus

(For reference, this is problem 10-15 in Lee's Introduction to Topological Manifolds, 2ed) We are given the annulus $$Q=\{z\in\mathbb{C}|1\le|z|\le 3\}$$ and the quotient corresponding to the ...
Nico's user avatar
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7 votes
0 answers
129 views

Different groups with the same classifying space.

Let $G$ be a topological group and $BG$ its classifying space. From the LES of the universal bundle, we get $\pi_i(BG)\cong\pi_{i-1}(G)$, so given the classifying space, we know all homotopy groups of ...
Reinhard's user avatar
7 votes
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453 views

Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point ...
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7 votes
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508 views

Good Reference for Large Cardinals/Homotopy Theory

I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
Jonathan Beardsley's user avatar
6 votes
0 answers
57 views

Is there a modern explanation of how Whitehead calculated $\pi_2^s$?

It was George W. Whitehead who first gave a correct calculation of the $2$-stem $\pi_2^s$ in The (n+2)nd Homotopy Group of the n-Sphere. The problem comes down to determining the suspension ...
Thorgott's user avatar
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6 votes
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129 views

Plus construction on Simplicial Sets?

Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that ...
wind's user avatar
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6 votes
0 answers
95 views

Do unit quaternions admit a continuous square root function?

Let $G$ be a multiplicative group. A function $\text{sqrt}: G \to G$ is a square root function iff, for every $x \in G$, $(\text{sqrt}(x))^2 = x$. The unit real numbers, namely $\pm 1$, doesn't even ...
Dannyu NDos's user avatar
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6 votes
0 answers
145 views

Motivation for Quasicategories

Recall that a simplicial set $X$ is called a quasicategory if for every $0<k<n$, every map $\Lambda^n_k\to X$ admits an extension to a map $\Delta^n\to X$. It is good to have a concise ...
Ken's user avatar
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6 votes
0 answers
245 views

Why do we care about $(\infty,2)$-categories?

Homotopy theory provides much motivation for studying $(\infty,1)$-categories in their relations to homotopical algebra, derived geometry, stable homotopy stuffs, cohomology, physics, and so on. As ...
Daniel Teixeira's user avatar
6 votes
0 answers
213 views

Cellular diagonal approximation

Let $X$ be a Co-H space with a finite CW structure. Composing the comultiplication $c:X \rightarrow X \vee X$ with the inclusion $i:X \vee X \rightarrow X \times X$ gives a map $$i \circ c \simeq \...
Matt's user avatar
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