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.
3,467
questions
0
votes
0answers
11 views
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 ...
1
vote
1answer
22 views
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}...
1
vote
1answer
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 $\space f(-x)=-f(x)$ $ \...
0
votes
0answers
20 views
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 ...
1
vote
0answers
24 views
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 ...
1
vote
1answer
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 ...
1
vote
1answer
26 views
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 $\exists$ ...
0
votes
1answer
52 views
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
$D^n$-...
0
votes
1answer
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 ...
1
vote
1answer
39 views
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 ...
0
votes
0answers
53 views
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 ...
-1
votes
1answer
39 views
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 $F : I \to X$ ...
0
votes
1answer
43 views
Homotopy equivalence among the following topological spaces
How do I show the homotopy equivalence among the following topological spaces:
0
votes
2answers
28 views
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 ...
3
votes
0answers
42 views
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. ...
1
vote
0answers
21 views
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 ...
0
votes
1answer
37 views
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 ...
-2
votes
1answer
59 views
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 ...
0
votes
0answers
24 views
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) \...
2
votes
0answers
54 views
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 ...
1
vote
1answer
26 views
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 ...
3
votes
2answers
102 views
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 ...
1
vote
0answers
19 views
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 ...
0
votes
1answer
38 views
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 ...
7
votes
1answer
108 views
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 ...
1
vote
1answer
49 views
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(...
3
votes
0answers
41 views
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 \...
2
votes
0answers
31 views
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 : ...
2
votes
1answer
37 views
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+...
6
votes
1answer
144 views
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 ...
0
votes
1answer
56 views
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(\...
2
votes
0answers
88 views
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 ...
1
vote
1answer
38 views
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 ...
3
votes
1answer
38 views
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 ...
0
votes
0answers
63 views
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? ...
0
votes
2answers
84 views
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 \...
1
vote
0answers
45 views
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 ...
0
votes
1answer
46 views
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,...
2
votes
0answers
51 views
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})=\...
0
votes
1answer
33 views
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 ...
2
votes
1answer
36 views
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 ...
1
vote
1answer
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-...
2
votes
1answer
65 views
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 NersƩs Aramian which can be found in this pdf.
It is essentially demonstrated that every path in $S^n$ is ...
2
votes
1answer
67 views
$E^*(\mathbb{C}P^{\infty})=\bigoplus_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$ 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 ...
1
vote
3answers
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 ...
1
vote
2answers
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 ...
1
vote
1answer
49 views
Isomorphism between homotopy groups of CW-complexes
Let $(Y, y_0)$ and $(Yā, y_0)$ pointed CW-complexes, with $Yā$ obtained from $Y$ by attaching $n+1$-cells. Why is it true that $i_{*}: \pi_{q}(Y, y_0) \to \pi_{q}(Yā, y_0)$ is an isomorphism for $q &...
1
vote
1answer
47 views
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 ...
2
votes
1answer
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 ...
2
votes
0answers
67 views
Integrating over homotopy classes
We know that $Öæ\pi_2(\mathbb{S}^2)\cong\mathbb{Z}$. 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 ...
