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

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Group homomorphism on unit circle

For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$. How would one compute the induced group ...
2
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0answers
17 views

What is the relationship between the path-loop space fibration and path induction?

I have an intuition that I can't quite put into words that the path-loop space fibration $\Omega X \rightarrow PX \rightarrow X$ and the based path induction axiom are related, but I don't know enough ...
5
votes
2answers
47 views

Homotopical perspective on the long exact sequence in homology and Mayer-Vietoris

For a while I have been wondering whether the long exact sequence in homology and the Mayer-Vietoris sequence can be phrased in homotopical terms. Recently, I heard that both may be reformulated via ...
3
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3answers
95 views

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map?

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map? I get the picture that if the identity map $id$ is homotopic to the constant map then as the circle transforms ...
2
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1answer
27 views

$|\beta(s)-\alpha(x)|<d$ implies that $\beta$ is homotopic to $\alpha$.

Let $D$ be an open subset in $\mathbb{R}^n$. Let $\alpha$ be a path in $D$ from $x$ to $y$, and set $d=\inf\{|\alpha(s)-w|:w\in \partial D, 0\le s\le 1\}$. Show that if $\beta$ is any path in $D$ ...
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1answer
36 views

Homotopy of a pair

Let $X$ be a topological space and $A \subset X$ a subspace. For $x \in A$, the homotopy group $\pi_n(X,A,x)$ of the pair $(X,A)$ is by definition the set of homotopy classes of n-cells relative to ...
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0answers
60 views

Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...
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1answer
35 views

Inclusions of CW-complexes are cofibrations.

Has the inclusion from the $ (n - 1) $-sphere in the $ n $-disc the left lifting property for all acyclic Serre fibrations? I am looking for a reference for this proposition, or alternatively, for an ...
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1answer
22 views

Homotopy between circle and semi-circle.

It is easy to see that a circle and a semi-circle (as in the image), of the same radius, are homotopic. I was trying to exhibit a homotopy, but it seems that I'll have to use analytic geometry, solve ...
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0answers
26 views

Where can I find Kan's paper “On c.s.s. categories” from 1957.

Does anyone know how I can find the following article by Daniel Kan: Kan, D. M. On c.s.s. categories Bull. Soc. Math. Mexicana (1957), 82-94. Quillen lists it as a reference in his paper Rational ...
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1answer
53 views

Derived functors - homotopical vs homological approach

In a first course in homological algebra, the lecturer introduced derived functors as universal $\delta$-functors, whose universal property is splicing short exact sequences into long ones. It so ...
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0answers
23 views

Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$.

Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$. If someone could verify my proof that would be great. I just started this learning this material and I ...
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1answer
27 views

Show that a circle with a point removed has the same homotopy type with a point.

Show that a circle with a point removed have the same homotopy type with a point. This is clear if we look at the picture, but I don't know how to actually write down a proof. I know that if ...
1
vote
1answer
86 views

Prove the long line is not contractible.

Given the following definition of the long line: Let $\omega_1$ be the first uncountable ordinal and consider $[0,1)$ as an ordinary set. Define the long ray to be the ordered set $\omega_1 \times ...
4
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0answers
31 views

Is the sphere with a diameter homotopy equivalent to a surface?

This is for a homework problem: Take the unit sphere $\mathbb{S}^2$ and join the north and south poles with a line segment. Is the resulting space homotopy equivalent to a surface? Intuitively, ...
3
votes
1answer
78 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
3
votes
1answer
42 views

almost complex structures on $R^4$

How should I see that the set of almost complex structures on $R^4$ preserving the positive orientation, namely $\{J\in GL^{+}(4,R), J^2=-I\}$ is homotopy equialent to $S^2$. There is a similar ...
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0answers
40 views

How to prove that the topological spaces are homeomorphic

Let's consider a topological space $X$. We define $X \times I / {\sim}$ as a product of $X$ and a $I=[0, 1]$ closed interval's quotient by the following equivalence relation: $(x, 1) \sim (y, 1), (x, ...
6
votes
4answers
198 views

What is an “inner isomorphism” between different groups?

It is well known that if $X$ is a path-connected topological space containing points $x$ and $y$, then the fundamental groups $\pi_1(X,x)$ and $\pi_1(X,y)$ are isomorphic. Wikipedia makes the further ...
3
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0answers
35 views

The relation between homotopy equivalence and contractible mapping cone?

In this MO thread, the OP claimed that it is obvious that homotopy equivalence implies the mapping cone contractible, whereas the converse proposition is wrong. I hate to admit that it's not obvious ...
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0answers
53 views

How do I show that this function is continuous? [duplicate]

I'm interested in showing that $CX=\frac{I\times X}{\{1\}\times X}$ is contractible. I defined the d.r $F(s,[t,x])=[(1-s)t+s,x]$ and the only missing part for me is to show that it is continuous. How ...
2
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0answers
41 views

Homotopy groups of mapping spaces

If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? ...
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0answers
57 views

“Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
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1answer
28 views

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$.

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$. The problem is in page 457 of "Topology and Geometry" written by Glen E. Bredon. I think I need to use a long exact sequence of ...
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1answer
20 views

Is the skeleton-coskeleton adjunction $sSet$-enriched?

Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?
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vote
1answer
44 views

Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
3
votes
0answers
74 views

Criterion for homotopy equivalence in the category of pair of spaces

I am trying to prove the following statement : Let $\{ * \} \subset A \subset B \subset X$ be a chain of topological spaces (all subsets have the subspace topology). It is given that $A \to X$ ...
0
votes
0answers
21 views

Long exact sequence of homotopy groups $\pi_n$ for a pointed homotopy pullback square

Let \begin{align} A &\to B\\ \downarrow &~~~~~\downarrow\\ C &\to D \end{align} be a homotopy pullback square of pointed simplicial sets. One gets a long exact sequence $$ ...
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votes
2answers
50 views

How to compute a homotopy to show the operation on the fundamental group is assoicative?

By definition $$[(\alpha *\beta) *\gamma ] (s) = \begin{cases}\alpha (4s) & 0 \leq s\leq \frac{1}{4} \\ \beta(4s-1) & \frac{1}{4}\leq s\leq \frac{1}{2}\\ \gamma(2s-1) & \frac{1}{2}\leq ...
3
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1answer
58 views

The Plate Trick and $SO(3)$

This is a followup of sorts to Plate trick demonstrating SO(3) not simply connected. . Suppose I do the ``plate trick'' to demonstrate the existence of an order-2 element in $\pi_1(SO(3))$. That is, ...
3
votes
1answer
35 views

Fundamental question about cohomology on the stable homotopy category

I've gotten myself tied in knots about this elementary derivation of a ludicrous conclusion. Appreciate a hand straightening myself out! (1) A fibration $F\to E \to B$ of CW complexes gives rise to a ...
2
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0answers
33 views

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
5
votes
1answer
58 views

Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over ...
0
votes
1answer
46 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
4
votes
0answers
33 views

Transversality and homotopic maps

I'm trying to solve some problems in differential topology, and I came across the following: suppose $f:M\times [0,1]\rightarrow N$ is a homotopy, where $M$ is a compact manifold, such that $f_0$ and ...
4
votes
1answer
107 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
6
votes
1answer
100 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
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votes
0answers
34 views

Question on cofiber sequence map in equivariant homotopy theory

Let $G=C_2$ denote the cyclic group with two elements. Up to isomorphism there are only two irreducible $C_2$-representations, the identity representation, $\mathbb{R}^{1,0}$, and the sign ...
1
vote
1answer
28 views

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph, and the question asks further "which graph" Let ...
0
votes
1answer
31 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
4
votes
1answer
61 views

Relationship between cohomology and higher-homotopy

Let $M$ be a connected, compact, and orientable 3-manifold ($H^3(M)\cong\mathbb{Z}$), and let $G$ be a simple Lie group satisfying $\pi_1(G)=\pi_2(G)=0$. Let $\pi_M(G)$ denote the set of homotopy ...
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vote
2answers
58 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
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vote
0answers
20 views

Tools for proving map is homotopy equivalence

General situation I'm preparing a geometry exam, and a lot of exercises from past years' exams are of the form «given $f$ the map so-and-so, prove (or determine whether) it is a homotopy ...
4
votes
3answers
80 views

The significance of filtered colimits in homotopy theory

I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask ...
0
votes
1answer
133 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
5
votes
1answer
118 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
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vote
1answer
47 views

Question about homotopic functions and homotopy classes

If $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$. I have to give an example of two continuous functions $f$ and $g$ that are not homotopic. I was thinking of $f(x,y) = x$ and $g(x,y) = y +2$ ...
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0answers
29 views

functors with a morphism lifting property

By analogy to the familiar situation in homotopy theory (i.e., (unique) path lifting in covering spaces), it is natural to consider the following. Let $P:C\to D$ be a functor. Say that $P$ has ...
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1answer
17 views

Relative homotopy and composition of maps

I am trying to prove something and am stuck on the following issue : Suppose $\Psi, \Phi : I^n \to Y$ are two maps and $q:Y \to Z$ is a homotopy equivalence such that $q \Phi \cong q \Psi $rel ...
2
votes
1answer
55 views

Find $f$ and $g$ homotopic s.t. induce different homomorphisms

Let $X$ and $Y$ be two topological spaces. Are there continuous functions $f,g:X\to Y$ satisfying the following conditions? $f(a)=g(a)=b$ for some $a\in X$, $f$ and $g$ are homotopic, and the ...