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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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Higher homotopy/homology groups of smash product of a space

Let $X$ be a finite CW complex and let $X \wedge X$ denote its smash product. Given the higher homotopy and homology groups $\pi_i(X)$ and $H_i(X)$, is there a simple way to calculate $\pi_i(X \wedge ...
ShamanR's user avatar
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On the boundary homomorphism of homotopy exact sequence

Denote $P^{n+1}(d)$ to be the mapping cone of the degree map $d:S^n\to S^n$, $q:P^{n+1}(d)\to S^{n+1}$ to be the pinch map and $F_q$ to be the homotopy fibre of $q$. In the exact sequence $$\pi_k(P^{n+...
Visible Wings's user avatar
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Fundamental group of creative intersecting circles in the space

Are my homotopic equivalences correct? I'm not sure about the contractions that I made, some suggestions?
Sigma Algebra's user avatar
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36 views

How to extend an ordinary vector bundle to a $\langle k \rangle$-vector bundle?

I have this question while I am studying Steimle's paper "An additivity theorem for cobordism categories". In this paper, a $\langle k \rangle$-vector bundle is defined to be a $P(\underline{...
Yuxun Sun's user avatar
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1 answer
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the infinite category constructed from a model category has all limits and colimits

I am reading a survey on derived algebraic geometry. On the page 28, I read such a paragraph: Let $C$ be a model category, and $I$ be a small category, let $C^{I}$ be the model category of diagrams in ...
Yang's user avatar
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1 answer
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Failure of universal coefficient theorem for spectra

Let $X, E $ be spectra and assume that the smash product $X\otimes E$ is nullhomotopic. Does it follow that the mapping spectrum $\underline{map}_{Sp}(X,E)$ is also null-homotopic; is every map $X\to ...
Fabio Neugebauer's user avatar
3 votes
1 answer
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Homeomorphism of $[0,1]^m$ with subspaces of $S^n$

I was reading the Borsuk-Ulam theorem, which states that there is no continuous map $f$ from $S^2$ to $S^1$ which satisfies $f(-x)=-f(x)$. One question came to my mind: Is there any subspace of $S^n$ ...
mahdi meisami's user avatar
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For any $\alpha \in \pi_6 {(\mathbb{C}P^2 \vee S^2)}$, $\mathbb{C}P^2 \vee S^2 \cup_{\alpha} D^7$ has the same cohomology algebra over a field.

In the book Rational Homotopy Theory by Y. Felix, S. Halperin, and J. Thomas. Exercise 4.3 asks the reader to prove that for any $\alpha \in \pi_6 {(\mathbb{C}P^2 \vee S^2)}$, the space $X_{\alpha}=\...
Jiahao Li's user avatar
2 votes
2 answers
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Homotopy equivalence between any Topological spaces.

I am beginning my studies in the area of homotopy theory, and I am studying from the book "Fundamental Group and Covering Spaces" by the Brazilian author Elon Lages Lima. As this is my first ...
Joel Marques's user avatar
3 votes
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How to understand the inverse loop in the fundamental group?

I'm struggling to build an intuitive picture of the inverse of a loop in the fundamental group. Is there an intuitive way to see that the if I start from a point $x$ of a topological space an travel ...
Emil Sinclair's user avatar
2 votes
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How is the $\pi_1(X)$ action on higher homotopy groups visible in $\Pi_\infty(X)$

It is well known that $\pi_1(X)$ acts on all the higher homotopy groups, and this action can be seen in several different ways see this question. I have recently started working with the fundamental ...
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Question about Deformation Retractions in Fomenko and Fuchs

Working through Anatoly Fomenko and Dmitry Fuchs' "Homotopical Topology". As an offhand comment, they mention: $A$ is a deformation retract of $X$ if and only if the inclusion map $A \...
Lukrau's user avatar
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How can one intuitively understand the dependence of the set relative to which homotopies are constructed?

In the notes I am using, homotopy is defined relative to a set. To start the discussion I present the definition: Two functions $f,g:(X, \tau_x) \to (Y,\tau_y)$ are said to be homotopic relative to a ...
Cathartic Encephalopathy's user avatar
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1 answer
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What are some applications of homotopy in topology other than constructing fundamental group (and higher homotopy groups)??

In pretty much all books I've seen, homotopy is introduced for the pure purposes of defining the fundamental group. But, what are some other of uses of homotopy? One that I found is that it can be ...
Cathartic Encephalopathy's user avatar
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1 answer
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What is homotopy relative to in homotopically equivalent topological spaces?

Two spaces $X$ and $Y$ are homotopy equivalent if there are continuous maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ with $gf \sim \textbf{1}_X$ and $fg \sim \textbf{1}_Y$. Before this section, ...
Cathartic Encephalopathy's user avatar
3 votes
1 answer
56 views

Composing sums of elements in $\pi_{n}(S^m)$

Suppose $m,n,r\geq 1$ and that we have homotopy group elements $f,g\in \pi_{n}(S^m)$ and $\alpha,\beta\in \pi_r(S^n)$. Certainly, it is the case that $f\circ (\alpha+\beta)=(f\circ\alpha)+(f\circ\beta)...
J.K.T.'s user avatar
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3 votes
1 answer
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Path-connected components and homotopy equivalence of topological space $\Delta$ of ordered triangles in $\mathbb{C}$.

I am having trouble with the following question, particularly the intuition behind it and visualizing the described space. Consider the subspace $\Delta$ of $\mathbb{C}^{3}$, of ordered triangles in $...
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How to know if certain path homotopy classes generate fundamental group

I am studying for a topology final and came across the following problem. Let $B^{2} = \{(x, y) \in \mathbb{R}^{2} \mid x^{2} + y^{2} \leq 1\}$ denote the unit ball. Consider the space $Y := B^{2} \...
JLGL's user avatar
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When is $\pi_k(X,A)\rightarrow \pi_k(X/A)$ an isomorphism

Let $A$ be a sub-CW-complex in $X$. A corollary of homotopy excision theorem states that if $(X,A)$ is $m$-connected and $A$ is $n$-connected, then the map $$\pi_i(X,A) \longrightarrow \pi_i(X/A)$$ ...
Akhalbing's user avatar
2 votes
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27 views

Relative homotopy equivalence despite not being a retract?

The following chain of implications, for a subspace $A$ of $X$, is well-known and easy: ($(X, A)\cong (Y, B)$ means that there exist $f\colon X\to Y$ and $g\colon Y\to X$ such that $f(A)\subseteq B$, ...
Atom's user avatar
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suspension map between homotopy groups of spheres

Let $p$ be an odd prime. For $n\geq 3$, suspension map $\Sigma:\pi_{n+2p-3}(S^n;p)\to\pi_{n+2p-2}(S^{n+1};p)$ is an isomorphism on $p$-primary part and $\pi_{2p}(S^3;p)=\mathbb{Z}/p$. Meanwhile, there ...
Visible Wings's user avatar
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Showing that the compositions $+_i$ in the fundamental double groupoid are well defined.

I'm trying to understand the fundamental double groupoid of a triple of spaces $ \rho_2(X,A,C)$ in the book Nonabelian Algebraic Topology and I'm having some confusion about the proof showing that the ...
Mu_Coffee's user avatar
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1 answer
35 views

Reference for a proof on homotopy groups of simplicial abelian groups

In Simplicial Homotopy Theory, Goerss and Jardine wrote : The simplicial abelian group structure on A induces an abelian group structure on the set $\pi_n$(A, 0) = [($\Delta^n$, $\partial \Delta^n$), ...
newuser's user avatar
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Simplicial presheaves present $\infty$-presheaves related question

I am trying to work out the details that every $\infty$-topos is presented by a model topos. By presented I mean it is the image under the homotopy coherent nerve. A model topos is a model category ...
Secher Nbiw's user avatar
1 vote
0 answers
14 views

Is homology distributive over addition of maps in a homotopy set?

Let $K$ be a co-H group with homotopy commutative comultiplication $\mu$ and $(X,x_0)$ be a pointed space. Denote $H_\ast$ to be the singular homology with integer coeffients. Take $f,g\in[K,X]$, does ...
Visible Wings's user avatar
1 vote
1 answer
60 views

Question about Whitehead Tower

I'm studying Miller lectures on Algebraic Topology, and I'm stuck in Theorem 68.9 ($\bmod \mathcal{C}$ Hurewicz Theorem ): Theorem 68.9 (Mod $\mathcal{C}$ Hurewicz theorem). Assume that $\mathcal{C}$ ...
marc's user avatar
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2 votes
2 answers
66 views

Naturality for the Homotopy Fiber Sequence of Mapping Spaces

For a cartesian fibration $p\colon\mathcal{E}\rightarrow\mathcal{C}$ of $\infty$-categories (quasicategories) and objects $x,y$ of $\mathcal{E}$, the induced map $\mathrm{map}_{\mathcal{E}}(x,y)\...
Thorgott's user avatar
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4 votes
0 answers
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Geometric fixed-points of MO

Recall that the value of the orthogonal spectrum $\mathbf{MO}$ at an inner product space $V$ is the Thom space of the tautological bundle over the Grassmannian of $|V|$-demensiomal planes in $V\oplus ...
yifan's user avatar
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Uniqueness of the octahedral axiom

The octahedral axiom for a triangulated category says that if I have two morphisms $f:A \to B$ and $g:B \to C$, then there is an exact triangle $Cf \to C(g \circ f) \to C(g)$ between the cones on $f, ...
categorically_stupid's user avatar
2 votes
1 answer
51 views

Is the inverse of an isotopy of embeddings continuous?

I was dealing with isotopies and monodromies when the following question arose, which is more subtle than it might seem at first sight. Let $X$ and $Y$ be topological spaces. Suppose that $\Phi: [0,1] ...
Don's user avatar
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1 answer
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Literature Search in Homotopy Theory

I am searching for a result in the literature which I found two months ago but did not properly record. I just wrote down the following information about it. Theorem (Hurewics 1939, Asphäresche Räume)....
Esteban Ricardo Castillo's user avatar
1 vote
2 answers
108 views

What is a simple intuitive example of a nullhomotopic map?

I'm just in the first week of an Algebraic Topology class and have been presented with the following definition. A map is nullhomotopic if it is homotopic to a constant map. I.e. $f: X \to Y$ is ...
JamesLevine's user avatar
1 vote
1 answer
39 views

Commutativity of a diagram involving braids

I have this diagram and I want to prove its commutativity Let me explain what it means. $\beta$ is a braid with $n$ strings, that is represented in $\mathbb{D} \times [0,1]$. Its uper ends are $\beta(...
Alejandro's user avatar
2 votes
1 answer
58 views

$\Sigma_n$ action on algebras of symmetric monoidal $\infty$-categories

Let $\mathcal{C}^\otimes$ be a symmetric monoidal ($\infty$)-category. In $1$-category theory, given $X^{\otimes n} \in \mathcal{C}$, there is an action on the permutation group $\Sigma_n \...
Frusciante's user avatar
3 votes
1 answer
70 views

Homotopy in $\mathbf{Top}_{\text{Quillen}}$.

Recall the classical model structure $\mathbf{Top}_{\text{Quillen}}$ on the category $\mathbf{Top}$ of (cgwh, if you want) topological spaces. In any model category, there are the notions of (left, ...
Thorgott's user avatar
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2 votes
1 answer
57 views

Multiplication map of algebras of monoidal ($\infty$)-categories

Lurie defines (DAG-II, Def. 1.1.2) a monoidal $\infty$-category $\mathcal{C}$ as a coCartesian fibration $\mathcal{C}^\otimes \to N(\Delta)^\text{op}$ satisfying the Segal condition, i.e. such that ...
Frusciante's user avatar
4 votes
0 answers
48 views

Higson's homotopy invariance result

I am learning about operator algebras and $KK$-theory, a result I find very striking is the following : Any split-exact $K$-stable functor $F : C^*\text{-alg} \to \text{Ab}$ is necessarily homotopy ...
Thil's user avatar
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Elementary proofs of the v1-periodicity of the Hopf map

I've been learning the periodicity theorem recently, and I know that the Hopf map $\eta \in \pi_1(\mathbb S)$ is $v_1$-periodic, which can be shown by machinery like ANSS. On the other hand, chromatic ...
Ziv's user avatar
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4 votes
3 answers
84 views

Does simply connectedness require fixing the endpoints when homotopying to a point?

A space $X$ is simply connected if it is path connected and every loop is homotopic to a constant loop. A space $X$ is simply connected if it is path connected and has trivial fundamental group. ...
hey's user avatar
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On Quillen’s Fiber Theorem

I am currently reading Vidit Nanda lectures notes on computational algebraic topology and come accross this version of Quillen's Theorem A : THEOREM 2.10. (Quillen's Fiber Theorem) Let $f: K \...
BabaUtah's user avatar
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27 views

Nerve theorem and cycles deformation

Consider a finite set of point $P\subset[0,1]^{d}$. Let $r>0$, by nerve theorem, we know that, the Čech complex $C^{2r}(P)$ and $B_{2}(P,r)$ are homotopy equivalent. Furthermore, Proposition 3.2 of ...
BabaUtah's user avatar
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51 views

2 functions are not isotopic

General question: In general, what methods do we have for showing 2 functions $f:X\to Y$ are not isotopic (i.e homotopic s.t each $H(*,t)$ is a homeomorphism$:X\to H(X,t)$ for every $t$) specific ...
Chris P. Bacon's user avatar
1 vote
0 answers
87 views

The notion of fundamental cogroup

I know there is a functor $\pi_1$ from the category of pointed topological spaces to the category of groups, sending each pointed topological space to its first fundamental group. I know that a group ...
toby flenderson's user avatar
0 votes
1 answer
28 views

Fundamental group of the cube skeleton minus a point

I'm trying to do Exercise 10.10 Compute the fundamental group of the space $X$ that is the union of the edges of the cube $C = [0, 1]^3$. Moreover, compute the fundamental group of $X \setminus \{x\}$...
hbghlyj's user avatar
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Homotopy between bounded linear operators on Banach spaces

In my course of Functional Analysis, talking about continuity method and index of Frendholm operators we construct homotopy between operators. However i encountered some difficulties to connecting ...
Manuel Bonanno's user avatar
3 votes
1 answer
75 views

Extensions by scalars along Frobenius

My question probably just needs the knowledge of some commutative algebra but it is ultimately motivated from the theory of formal groups in chromatic homotopy theory. Question. Let $R$ be an $\mathbb{...
Qi Zhu's user avatar
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1 vote
1 answer
50 views

How is homotopy lifting related to the general lifting problem?

I am getting re-introduced to Algebraic Topology. To motivate things up, the lifting problem was stated: Given a map $p\colon E\to B$ and a map $f\colon X\to E$, does there exist a map making the ...
Atom's user avatar
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5 votes
1 answer
103 views

Higher Coherences and Maps from Colimits

In higher category theory there is the important mantra that commutativities are additional data and not just a property as in $1$-category theory. So the following question I'm proposing will ...
Qi Zhu's user avatar
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5 votes
1 answer
130 views

Natural transformation picking out the map from the initial object

As so often, I'm failing to construct a map in $\infty$-category theory that is easily constructed in $1$-category theory. Let $\mathscr{C}$ be an $\infty$-category and let $F: \mathscr{C} \to \mathsf{...
Qi Zhu's user avatar
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Reference Request for Comparison Theorem of Motivic Cohomology

I encountered the following theorem: Let $X$ be a smooth scheme over a field $k$, then whenever $p \ge 2q-1$, we have an isomorphism $$ H^{p,q}(X,\mathbb Z)\cong H^{p-q}(X, K_q^M) $$ where $K_q^M$ is ...
SummerAtlas's user avatar
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