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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Free homotopy of a closed curve must be a closed curve

I'm reading a textbook on differential geometry and the author proceeds to define a weaker version of homotopy. Two continuous curves $c_0, c_1: [a,b] \to U \subseteq \mathbb{R}^n$ with the same ...
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Homotopy-coherent replacement of diagrams in quasi-categories

In Emily Riehl's book Categorical Homotopy Theory, a proposition (16.3.1) attributed to Cordier-Porter states that if $\underline{\mathcal{C}}$ is a fibrant simplicial category, and if $F : \mathcal{A}...
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39 views

$hf\alpha:I\to S^1$ is not null-homotopic

I was studying this proof, I understood everything, except the part where $hf\alpha:I\to S^1$ is not null-homotopic. We have, that $f:S^2 \to S^1$ a continous map such that $\space f(-x)=-f(x)$ $ \...
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homotopy and arrival space of an homotopic function

I have an exercise to solve where the goal is to show that a space $X$ is homotopic equivalent to a point if and only if, $\forall f:X\to Y$ is nullhomotopic for any given $Y$. I think that the best ...
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For a fibration $p:E\rightarrow B$, if $b,b'\in B$ are in the same path component, then $F_b\simeq F_{b'}$

If $p:E\rightarrow B$ is a fibration (Hurewicz), and we take $b,b'\in B$ in the same path component, I want to prove that the fibers are of the same homotopy type $F_b\simeq F_{b'}$. I know I should ...
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45 views

Homotopy fixed points and ordinary fixed points

Currently I know nothing about homotopy fixed points except for its definition: given a $G$-space $X$, the set of homotopy fixed points is defined as the space of equivariant maps from $EG$ to $X$. I ...
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Difference between homotopy function and homotopic space

I'm having some troubles understanding the concept of homotopy and homotopic spaces. I understood that given two function $f,g:X\to Y$ we can say that there's an homotopy from $f$ to $g$ if $\exists$ ...
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Question on Homotopy and Homology Groups of Spaces $X$ and $X \cup_{S^{n-1}} D^n$

I have a general question about the differnce of homology and (simplicial) homotopy groups of a topological space in degrees $X$ and the space $Y := X \cup_{S^{n-1}} D^n$ obtained by glueing $D^n$-...
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43 views

Is it possible that a surjective map on a non contractible space be null-homotopic?

Let $X$ be a non contractible space and $f:Y\to X$ a surjective map. Is it possible that one have $f$ null-homotopic? If yes, Is there some result to help to prove that some specific surjective map ...
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How do I show that a domain is not simply connected?

Let $U$ be a star-shaped domain in $\mathbb{C}$. Prove that the subset obtained from $U$ by removing a finite number of points is not simply connected. Usually when we want to show something is ...
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On Homotopy Equivalence

If a space X is homotopic equivalent to Y and X is contractible, prove that Y is contractible I know that since X is contractible the identity map is nullhomotopic and since X,Y are homotopic ...
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Homotopy between maps, Rotman Algebraic topology

Let $x_0,x_1$ belong to $X$ and let $f_i : X \to X$ for $i=0,1$ denote the constant map at $x_i$. Prove that $f_0$ and $f_1$ are homotopic if and only if there is a continuous function $F : I \to X$ ...
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Homotopy equivalence among the following topological spaces

How do I show the homotopy equivalence among the following topological spaces:
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Understanding why path $p*p^{-1}$ is homotopic to the identity/trivial loop

I saw this question, but the top answer has the same picture that is confusing me currently, so I was wondering if somebody could explain it to me. I'm reading Concepts of Modern Mathematics by Ian ...
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Representable cohomology theories in motivic homotopy theory

I am reading Mazza, Voevodskys and Weibels book on Lecture Notes on Motivic Cohomology and have grown curious about the following question: Which cohomology theories on $Sm/k$ is representable, i.e. ...
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Show that the pair $(X,[-1,1])$ and $(Z,[-i,i])$ satisfy the Homotopy Extension Property.

(Propoisition 0.17 - Hatcher) If the pair $(X,A)$ satisfies the homotopy extension property (HEP) and $A$ is contractible, then the quotient map $q: B \rightarrow B/A$ is a homotopy equivalence. A ...
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Continuous functions $f$, $g$ from $X$ to $\mathbb{R}^n\setminus \left\{0\right\}$ are homotopic

Suppose we have $f$, $g$ continuous mappings from a space $X$ to $\mathbb{R}^n\setminus \left\{0\right\}$ with $\|f(x)-g(x)\|\leq \|f(x)\|$, prove that $f$, $g$ are homotopic. Is anyone willing to ...
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What is the simplest object in mathematical space, other than a point in Homotopy theory?

Background An Intuitive Introduction to Motivic Homotopy Theory - Vladimir Voevodsky In this video at 2:44, Vladimir Voveodsky start its presentation of spaces with a point and say that there is one ...
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Long exact sequence of homotopy groups groups in simplicial sets - reference request

I believe it is well-known that for a based map $f:X\to Y$ of simplicial sets (possibly with some extra hypotheses on $X$ and $Y$), there is a long exact sequence $$ \ldots \to \pi_n(F)\to \pi_n(X) \...
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Need help understanding the existence of a projection map $pr_2:M \wedge X_+ \rightarrow X$

I am reading these introductory notes to cobordism theory, however cobordism theory is not required to understand my question. I just need help understanding a certain map. On page number 8 the author ...
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Are the lifts of freely homotopic closed paths (with different start/endpoints) also closed freely homotopic paths?

Weaker versions of this question have been asked before, but I am wondering about some more general cases: Suppose that $\pi: E \to B$ is a covering space and $\alpha, \beta: [0, 1] \to B$ are closed ...
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Homotopy groups of non-contractible manifolds

Motivated by a proof in a differential geometry book and so far my lack of knowledge in algebraic topology I would like to know the following : Is it possible to have a compact non-contractible ...
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Show two RCPCs are regularly homotopic

Show that the following regular parametrised closed curve in $\mathbb{R}^3$, defined on the interval $[0, 1]$, are regularly homotopic. Construct a regular homotopy between them and verify that it ...
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Is the cylinder homotopic to the circle without using real multiplication?

Let $X$ be a nice topological space (for example $X = \mathbb{S}^1$). It is possible to show by hand that the cylinder $X \times [0,1]$ is homotopic to $X$. All the proof I know rely on the ...
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non-trivial finite fundamental group

I am a physics student and I want to know about finite fundamental group of subspace of $\mathbb{R}^3 $. Is there a subspace of $\mathbb{R}^3$ with nontrivial finite fundamental group? I am just ...
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Action of $\pi_1(G)$ on $\pi_n(G)$ is trivial for a topological group $G$, i.e. $G$ is a n-simple space.

My question is a follow up of this question Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$. For a topological group $G$, there is a natural action of $\pi_1(...
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Homeomorphism and Homotopy equivaence among 3 circles with 3, 2 and 1 common point.

I need some help with the the following problem. I am not aware what technique should I use to solve the problem: In $\mathbb{R}^2$, we denote by $S_{x,y}$ a unit circle with center at $(x,y) \in \...
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Let $p_n$ : $S^1$ -> $S^1$ be the covering map $p_n$(z) = z, (complex notation), $n \in \mathbb{Z}$, n not $0$.

Prove there is a lift of $p_m$ through $p_n$ if and only if m=nk, for some $k \in \mathbb{Z}$. Clearly we have to use the Theorem that says: there exists a lift of $f$ (that is, a continuous map $g : ...
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Vanishing of $\pi_n(X\times Y, X\vee Y)$ for $n\leq p+q+1$

Let $X,Y$ be (based) CW-complexes (assume $X,Y$ are nice so that there is no problem with $X\times Y$). If $X$ is $p$-connected and $Y$ is $q$-connected, how to show that $(X\times Y, X\vee Y)$ is $(p+...
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Is the splitting $\pi_{k}(X,A)\simeq\pi_{k}(X)\times \pi_{k-1}(A)$ a $\pi_1(A)$-modules isomorphism?

Let $(X,A)$ be a pair of topological spaces with $A\subset X$. Fix a basepoint $x_0$ of $X$ which lies in $A$. Assume that the inclusion $(A,x_0)\to (X,x_0)$ is homotopic to a constant map relatively ...
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Are nullhomotopic maps trivial in $Set$?

Let $F$ a contravariant functor $F: Ho(Top) \to Set$. Let $f$ a morphism in $Ho(Top)$ such that $f$ is nullhomotopic, which means that $F(f) = F(c)$, where $c$ is the constant map. Knowing that $F(\...
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how to prove the existence of solution of system of equations using the degree of Brouwer?

Help me! Using the theorem (Existence and uniqueness of the Brouwer degree). There is a unique map $\deg_{B}: \Sigma \rightarrow \mathbb{Z}$ (Brouwer degree), where $\mathbb{Z}$ represents the set of ...
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Definition of the ring of weakly modular forms over $\mathbb{Z}_{(p)}$

I am an undergraduate in a small mathematics course focused on a single project. Right now we are in the preliminary stage, the part where our advisor has given us a high-level reference, referred us ...
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Does the category of semisimplicial Kan complexes form a category of fibrant objects?

It is known that the category of Kan complexes form a category of fibrant objects, as remarked here. We obtain obvious notions for fibrations and weak equivalences in the category of semisimplicial ...
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Concrete constructions in derived algebraic geometry

I am trying to understand basic constructions in derived algebraic geometry (such as derived Hilbert schemes or the derived stack of vector bundles). What is a good reference for learning about these? ...
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Is $\mathbb{S}^3 \backslash \mathbb{S}^1$ homotopic equivalent to $\mathbb{S}^1$?

Is $\mathbb{S}^3 \backslash \mathbb{S}^1$ homotopic equivalent to $\mathbb{S}^1$? I am reading this from some notes on algebraic topology. I am not even sure what does $\mathbb{S}^3 \backslash \...
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Homotopy extension property with maps at both ends fixed.

Consider the following property of a map $\iota : A \to X$. Given any pair of maps $f, g : X \to Z$ and a homotopy $H : A \times I \to Z$ between $f \circ \iota$ and $g \circ \iota$ it can be extended ...
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Homotopy equivalence between pointed CW-complexes

Let $f : (X, x_0) \to (Y, y_0)$ a morphism in the category of pointed, connected CW-complexes. Let $A= [1/2, 1]^{+} \wedge X \cup Y \subset C_f$, where $C_f$ is the mapping cone and $X, Y$ are pointed,...
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How do I understand the ratios between $\pi_1(U(N))=\mathbb{Z}$, $\pi_1(U(1))=\mathbb{Z}$, and $\pi_1(PSU(N))=\mathbb{Z}/N$?

We know that $$\pi_1(SU(N))=0, \tag{1}$$ $$\pi_1(PSU(N))=\pi_1(SU(N)/(\mathbb{Z}/N))=\pi_0(\mathbb{Z}/N)=\mathbb{Z}/N.\tag{2}$$ Also that $$\pi_1(U(N))=\pi_1(\frac{SU(N)\times U(1)}{\mathbb{Z}/N})=\...
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Degree of a smooth proper function, smooth homotopy and their pullbacks

Let $M$ and $N$ be compact manifolds, and $F, G: M \rightarrow N$ two smooth maps such that $F \sim G$, where $\sim$ means smooth homotopic. Then a well-known result is that the degree of $F$ is the ...
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Homotopy push-out squares and exact triangles are colimits

I read somewhere that homotopy push-out squares and exact triangles in a triangulated category can both be interpreted as special cases of higher categorical colimits. Why is this true? Please note ...
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62 views

Equivalent versions of the Mayer-Vietoris axiom in Brown theorem

In the hypotheses of Brown representability theorem there is a contravariant functor F from pointed connected CW complexes to pointed sets, which must respect two axioms, the second of which is the so-...
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Every path in $S^n$ is homotopic to a non-surjective path

I want to prove that the fundamental group of $S^n$ is trivial. I found this demonstration by Nersés Aramian which can be found in this pdf. It is essentially demonstrated that every path in $S^n$ is ...
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$E^*(\mathbb{C}P^{\infty})=\bigoplus_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$ or $\prod_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$?

PRELIMINARY DEFINITIONS: Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$ \tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt) $$ So there is a special ...
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67 views

Examples of continuous functions that are not homotopic to each other.

Let $f,g:X \to Y$ be continuous functions. Then, consider $F:X \times I \to Y$ defined as follows:$$F(x,t) = f(1-t)+gt$$ How does one check whether $F$ is continuous(w.r.t the product topology)? Is ...
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47 views

Correspondance between section of covering space and points in the preimage

Let $p:\tilde{X}\to X$ be a covering map (where $X$ is path conncetd and locally path connected). We call a continuous map $s:X\to\tilde{X}$ a section if $p\circ s=id_X$. Fix a point $x_0\in X$. Prove ...
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49 views

Isomorphism between homotopy groups of CW-complexes

Let $(Y, y_0)$ and $(Y’, y_0)$ pointed CW-complexes, with $Y’$ obtained from $Y$ by attaching $n+1$-cells. Why is it true that $i_{*}: \pi_{q}(Y, y_0) \to \pi_{q}(Y’, y_0)$ is an isomorphism for $q &...
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Simple exercise in fundamental groups and identification degree

I know this is simple, but I'm not being able to do this. a) Let $\Phi : \mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1$ be the multiplication of complex numbers, $\Phi (z_1, z_2) = z_1z_2$ and let ...
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37 views

Pushouts in category of topological spaces modulo homotopy

Let $HTOP$ denote the category whose objects are topological spaces and whose morphisms are equivalence classes of continuous maps modulo homotopy equivalence. I m wondering what are the properties ...
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67 views

Integrating over homotopy classes

We know that $ֿ\pi_2(\mathbb{S}^2)\cong\mathbb{Z}$. Given a fixed degree, say $n\in\mathbb{Z}$, is there a standard, conventional way to parametrize continuous maps $\mathbb{S}^2\to\mathbb{S}^2$ which ...

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