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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7
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142 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 ...
1
vote
2answers
68 views

Need help with computing homology group.

Let $D=$$S^2\cup$ x-axis$\cup$ y-axis be surface in $R^3$ I want to compute the homology group $H_n(D,\mathbb{Z})$ forcannot all $n\geq 0$ using Mayer-Vietoris Exact sequence. There exists many open ...
0
votes
0answers
47 views

Covering Manifolds

Let M a 3-manifold not orientable with incompressible boundary toral (possibly empty) and N the 2-sheeted cover space orientable of M. Questions: a) Why N has boundary toral(possibly ...
9
votes
1answer
418 views

Is the quotient map a homotopy equivalence?

It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy ...
3
votes
0answers
164 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
2
votes
1answer
117 views

negative Euler characteristic $\Rightarrow$ homotopy unique up to homotopy

In a paper by John Franks I stumbled upon the following: Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is ...
3
votes
2answers
130 views

Do pushouts of compactly generated Hausdorff spaces exist?

Let $A\to X$ and $A\to Y$ be maps of compactly generated Hausdorff spaces. Does the pushout $X\coprod_A Y$ in the category of compactly generated Hausdorff spaces exist? If necessary, one can assume ...
0
votes
1answer
115 views

Chain homotopy and compositions of morphisms.

Show that if $\alpha_1 \sim \beta_1$ and $\alpha_2 \sim \beta_2$ , then (whenever composition makes sense) $\alpha_1 \circ \alpha_2 \sim \beta_1 \circ \beta_2$. I have two questions. So are these ...
5
votes
1answer
62 views

local gauge invariance of field's homotopy class? Every map $S^2\rightarrow \mathrm{group } G$ is homotopic to a constant map?

In a discussion of a gauge field theory with gauge group $G$, someone says we can use a celebrated result of E. Cartan to show the gauge invariance of matter field's homotopy class. And Cartan's ...
2
votes
1answer
66 views

Spaces sharing all higher homotopy groups

Is it possible that two topological spaces share all higher homotopy groups, but are not homeomorphic? I should note that I have not studied much in the way of the theory of higher homotopy groups; I ...
0
votes
1answer
45 views

symmetry in the homotopy relation

Suppose $\alpha, \beta : I \to X$ are paths and suppose $\alpha $ is homotopic to $\beta$, $\alpha \cong \beta$. So, can find a continuous function $F(s,t) = f_t(s)$ such that $$ f_t(0) = \alpha(0) ...
7
votes
1answer
186 views

What exactly is duality?

In general, I am familiar with this notion of duality (i.e. in category theory, a statement is dualized simply by "reversing all arrows" and leaving objects unchanged). There are a couple of questions ...
2
votes
3answers
93 views

Symmetry of “is homotopic to” detail in the proof

Let $f,g:X\rightarrow Y$. If $f$ is homotopic to $g$ then $g$ is homotopic to $f$. Let $F:X\times I\rightarrow Y$ be a homotopy from $f$ to $g$ so $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x \in ...
1
vote
3answers
105 views

Number of homotopy classes

For topological spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of continuous maps $X\to Y$. If $I=[0,1]$ is the unit interval, then $[X,I]$ has only one element. If $X$ is path ...
1
vote
1answer
54 views

How to show that homotopy is preserved after composition?

I have two homotopies: $f\simeq f'$ and $y\simeq y'$. How can I show that $fy\simeq f'y'$ is again a homotopy?
2
votes
0answers
36 views

2 out of 3 axiom and simplicial sets

Let $i\colon\mathcal W\to\mathcal C$ be the inclusion of a subcategory. Unless I'm mistaken, the 2 out of 3 axiom for $\mathcal W$ to be a category of weak equivalences can be expressed as the ...
0
votes
1answer
179 views

Isomorphism functorially

I was reading the lecture notes of Pierre Schapira http://www.math.jussieu.fr/~schapira/lectnotes/AlTo.pdf I am not able to understand one thing. Please help. In page 75, theorem 4.6.1, the author ...
2
votes
0answers
51 views

Clarification about a double delooped H-space.

I've just started reading J.P. May's book The Geometry of Iterated Loop Spaces and am misunderstanding something. Somewhere, it's asserted that if an H-space X can be delooped twice, the its ...
1
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0answers
84 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
1
vote
1answer
53 views

Maps between surfaces of genus n

Could anyone help me find maps $f:\Sigma_{n}\longrightarrow\Sigma_{m}$ with $0<n<m\ \ $ NOT homotopic to a constant map or prove its existence without using K-theory? Thanks
0
votes
1answer
77 views

Question on “Homotopy invariance”

i have this from Hatcher's book "Algebric topology" And i don't understand why $\displaystyle \partial P(\sigma)=\sum_{j\leq i}(-1)^i(-1)^j F\circ (\sigma\times ...
0
votes
2answers
58 views

Homotopy between $X\setminus\{x\}$ and $A$

Let $X$ be a space obtained from $A$ by attaching a n-cell $e$. If $x\in e$ how to prove that the inclusion map $i_{A\to X\setminus\{x\}}: A\to X\setminus\{x\}$ is a homotopy equivalence?
5
votes
3answers
411 views

mapping homotopic to the identity map

Please give me a hand with this problem, It was on my exam, and I just couldn't solve it. Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is ...
6
votes
1answer
219 views

Tate Circles in Motivic Homotopy Theory

First off, this is a vague question about a survey which is, I guess, meant to be vague. So bear with me In Morel's "Motivic Homotopy Theory" survey he mentioned the following fact in motivating the ...
2
votes
0answers
78 views

About the universal bundle $EG\rightarrow BG$

For a topological group $G$, we define $EG$ to be the infinite join of $G$, and $B$ to be the quotient of $EG$ by the left action of $G$. Explicitly $EG$ can be expressed, as a set, as ...
5
votes
2answers
100 views

Map induced by $O(n)\hookrightarrow U(n)$ on homotopy groups

There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following ...
1
vote
1answer
99 views

Simply connected subset of $\mathbb R^3$

Let $C$ be the closed unit cube in $\mathbb R^3$, and let $A$ be one face of the cube $C$ (say the face above and parallel to $xy$-plane). Let $U\subset\mathbb R^2$ be open and path-connected such ...
1
vote
0answers
36 views

Finding a homotopy map

Let $K=\mathbb R^2\times (-\infty,0)\subset \mathbb R^3$, and let $Q$ be an open connected subset of $\mathbb R^2$. Is the fundamental group $\pi_1(Q\times [0,1)\cup K)$ trivial? And is it possible ...
2
votes
1answer
65 views

Collapse of a subspace - Cofibration

Let $i:A \rightarrow X$ be a (closed ) cofibration (i.e a cofibration in the Strøm Model structure). For a subspace $B \subset A \subset X$, when is it true that $A/B \rightarrow X / B$ is a ...
0
votes
2answers
83 views

Proof that two homotopy inverses are homotopic

Let $X$ and $Y$ be topological spaces. A continuous mapping $f : X \to Y$ is said to be a homotopy equivalence if there exists $g : Y \to X$ continuous such that $g\circ f$ is homotopic to $id_{X}$ ...
3
votes
0answers
243 views

Want to show two maps are homotopic

I am trying to solve the following problem but so far I cannot do it. Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group. Let $M$ be a closed manifold ...
7
votes
3answers
475 views

Is there any example of space not having the homotopy type of a CW-complex?

What is an example of space not having the homotopy type of a CW-complex? Is there any general method that can prove that the given space does not have the homotopy type of a CW-complex? (added) It ...
6
votes
1answer
361 views

Where to get help with Homotopy type theory?

I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ...
5
votes
0answers
75 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
0
votes
1answer
48 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
2
votes
1answer
169 views

Intuition behind a retraction from the cylinder onto the mapping cylinder.

Please excuse me for including pictures, but I thought it was easier than trying to redraw them here. I am right now reading Strøm's book Modern Classical Homotopy Theory. I have encountered a ...
5
votes
3answers
799 views

A confusion about the fact that contractible spaces are simply connected

Question 1: Greenberg's Algebraic topology has a proof that contractible spaces are simply connected. In the middle of the proof, the book makes use of the following fact without justifying it ...
2
votes
3answers
271 views

A question about the proof of the fact that contractible spaces are simply connected

In greeberg's algebraic topology, the following fact is used in the proof that contractible spaces are simply connected without justification: Let $p:\mathbb{I}\rightarrow X$ be a continuous function ...
2
votes
2answers
134 views

Question on homotopy

What is the relation between the definition of homotopy of two functions " a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined ...
4
votes
2answers
285 views

Intuition behind definition of homotopic equivalence and distinction with homeomorphism

I am a physics student and have come across the definition of homotopic equivalence of two spaces as existence of two functions $f:X \to Y,g: Y \to X$ such that $g \circ f$ and $f \circ g$ are ...
2
votes
1answer
287 views

Meaning of Fundamental group of a graph

I am a computer science student working in graph algorithms. I am unable to understand what the fundamental group of a graph means. I have some intuition regarding the fundamental group of a ...
11
votes
3answers
405 views

Showing continuity of partially defined map

There is a theorem in Note on Cofibrations by Arne Strøm. It says Let $A$ be a closed subspace of a topological space $X$. Then $(X,A)$ has the HEP if and only if there are (i) a neighborhood ...
4
votes
1answer
158 views

Bijection between homotopy classes and basepoint-preserving homotopy classes

$[X,Y]$ is the homotopy classes of maps from $X$ to $Y$ and $[X,Y]_0$ is the based homotopy classes of based maps. If $Y$ is path-connected and $\pi_1(Y)$ is abelian, then is the inclusion $$[X,Y]_0 ...
1
vote
2answers
40 views

Seeking 'simple' space with specified homotopy

I am looking for a 'named' space $S$ such that $\pi_1(S) = \mathbb{Z}_2$ and $\pi_n(S) = \star$ (the one-point group) for all $n\geq 2$. Commentary: I know that the projective plane fits the first ...
7
votes
1answer
127 views

Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) ...
2
votes
1answer
291 views

Difference between free homotopy and isotopy. Numer of non-isotopic curves.

I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the ...
1
vote
1answer
60 views

Is the continuous map between CW-complexes a cofibration?

If $f:A \rightarrow X$ is a continuous map between CW-complexes, then is $f$ necessarily a cofibration? I know that when $A$ is a subcomplex of $X$ and $f$ is the inclusion, the conclusion is true. ...
2
votes
1answer
84 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
4
votes
2answers
163 views

Can abstract nonsense be helpful here?

Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top ...
3
votes
1answer
61 views

Path homotopy in the plane

Let $C$ be a closed and simply connected subspace of the Euclidean plane $\mathbb{R}^2$. Suppose we have two simple paths in $C$, continuous functions $\alpha, \beta : [0,1] \to \partial C$, and ...