0
votes
1answer
45 views

Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ? I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point ...
1
vote
0answers
13 views

Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
0
votes
3answers
60 views

Homotopy on the unit circle

I am trying understand why the identity function on the unit circle $X=\{(x,y): x^2+y^2=1\}$ is not homotopic to $f: X \to X$ where $f(z)=(1,0)$ for all $z\in X$.
0
votes
1answer
29 views

problems with proving that f and g are homotopic.

i need to give an example of 2 continuous functions $f,g: X \rightarrow Y$ which are not homotopic, with: $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$ and i need to show how many homotopical ...
0
votes
1answer
55 views

Fundamental Group equaling 0

Let $X$ be a space for which $\pi(X,x)=0$. If $f,g$ are two paths in $X$ with $f(0)=g(0)=x$ and $f(1)=g(1)$, why is $f$ equivalent to $g$?
0
votes
2answers
40 views

Proving homotopy of paths

Let $f$ be a path in $X$ and $h:[0,1] \mapsto [0,1]$ a continuous mapping with $h(0)=0$ and $h(1)=1$. How can I prove that $f$ and $fh$ are homotopic relative to the endpoints?
3
votes
2answers
164 views

Why is $\pi_1(X,x_0)$ a group?

I want to show that $\pi_1(X,x_0)$ is a group. I am told that $e(t) := x_0$ is the identity element. Now, I am struggling to show that it is an identity element, and also that the inverse of an ...
3
votes
0answers
41 views

Path-homotopic definition.

Given two paths $f,g: [0,1] \mapsto X $ Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} ...
0
votes
1answer
27 views

Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
2
votes
1answer
44 views

Example for a space that is contractible to precisely one of its points

Give an example for a space that is contractible to one of its points and is not contractible to another of its points. I am really curious about that space, I have thought about tree with $n$ ...
0
votes
2answers
56 views

Quasi circle is not contractible

I'm trying to show that the quasi circle (picture below) doesn't have the homotopy type of a CW complex. I proved that all homotopy groups are zero. Now I need to show that it is not contractible to ...
3
votes
3answers
134 views

how should I show that it is wedge of infinite circles?

I know that the shape that we see it below is homotopy equivalent of wedge of infinite circles,so the fundamental group of it is $\prod _{1}(\vee _{\alpha \in A}S^{1})=\ast _{\alpha \in A ...
1
vote
0answers
14 views

Terminology for homotopies which stay inside some finite stage of a union

Sometimes it happens that you have a sequence of topological spaces each contained in the next $$ X_1 \subset X_2 \subset X_3 \subset \ldots$$ and you want to talk about things like homotopy in the ...
4
votes
1answer
87 views

Topology of space of continuous functions

Let $X,Y$ be topological spaces and let $C^0(X,Y)$ be the set of continuous functions between them, endowed with the compact-open topology. I am interested in the following kind of questions: What ...
3
votes
1answer
107 views

Relation between the braid group and the mapping class group of the plane

According to the following link, page 248, the braid group modulo its center is isomorphic to the mapping class group of the $N$-times punctured plane, i.e. $B_N/Z(B_N)\cong M_N(\mathcal(R)^2)$. Could ...
3
votes
2answers
63 views

Continuity of homotopy from constant function to identity

According to the definition of homotopy between two maps $f,g : X \to Y$ we need a continuous map $F : X \times [0,1] \to Y$ such that $F(x,0) = f(x)$ and $F(x,1) = g(x)$. Most examples I've seen in ...
4
votes
2answers
61 views

Non-closed deformation retract

I am searching for a $T_1$ space $X$ which deformation retracts onto a non-closed subspace $A$. Such a space cannot be Hausdorff as any retract in a Hausdorff space is closed. I tried some spaces, ...
2
votes
1answer
130 views

Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell

Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent. I think it can be proved by showing they are both deformation retracts of ...
9
votes
1answer
75 views

Is a bijective homotopy equivalence with bijective homotopy inverse a homeomorphism?

I've been thinking about this for a while, but didn't get very far. Maybe someone here can say something about it. I know of an example of two spaces $X, Y$ with continuous bijections in both ...
2
votes
1answer
79 views

Union of 2-sphere with line segment in $\mathbb{R}^3$ removing one point homotopy equivalence.

I am working on a problem from Lee's Introduction to Topological Manifolds where one is asked to compute fundamental groups using Van Kampen's theorem. I know how to use Van-Kampen's theorem but I ...
2
votes
1answer
67 views

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open ...
3
votes
2answers
75 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 ...
1
vote
1answer
45 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?
5
votes
3answers
151 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 ...
2
votes
0answers
54 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 ...
0
votes
2answers
45 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 $id_{X}$ ...
5
votes
3answers
170 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 ...
2
votes
1answer
105 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 ...
1
vote
2answers
76 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 ...
10
votes
3answers
369 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 ...
1
vote
1answer
43 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. ...
3
votes
1answer
46 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 ...
2
votes
1answer
74 views

Show that two different embeddings of the figure-eight in the torus are not homotopic

Note, we can express the torus $|X.| \cong T$ as a square with edges denoted by $e$ and $f$, the diagonal by $g$, and faces $T_1$ and $T_2$, and a single vertex $v$, with appropriate identifications. ...
5
votes
2answers
194 views

Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?

It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is necessary to make $\left(X,A\right)$ a pair having the homotopy ...
5
votes
3answers
344 views

Statement about Homotopy in Brown's “Topology & Groupoids”

I am trying to understand a statement in Brown's Topology and Groupoids, 7.2.5 (Corollary 1), page 270. Let's first have some preliminary remarks Let $X,Y$ be topological spaces. The track groupoid ...
0
votes
2answers
361 views

Deformation retract and homotopy equivalence

If $A\subset X$ is a deformation retract of $X$. Are $X$ and $A$ homotopy equivalent?
3
votes
1answer
139 views

Homotopy problem for infinite dimensional topological space III

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a $n$-dimensional regular CW complex. ...
0
votes
1answer
35 views

Deformation retract needs to be smooth?

So I am not quite sure that why none of these three is a deformation retract - is that because of the corners? But I don't remember deformation retract rely on smooth criteria, instead, on continuous ...
0
votes
1answer
93 views

Equivalence of path-connected CW-complexes and CW-complexes with one 0-cell

Proposition Any path-connected CW-complex is homotopy equivalent to a CW-complex with precisely one 0-cell. Proof (Sketch) Let $X$ be a path-connected CW-complex, so $sk_1(X)$ is a connected graph. ...
1
vote
1answer
48 views

Is a weakly contractible connected metric space, uniquely geodesic?

A topological space is weakly contractible if all the homotopy groups are trivial. It's connected if it's not the union of two disjoint nonempty open sets. A metric space $(X,d)$ is uniquely geodesic ...
3
votes
0answers
53 views

Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman. Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and ...
1
vote
1answer
93 views

Loop spaces and filtered colimits

I have read many things now that lead me to believe that the loop space functor preserves filtered (and/or directed) colimits. Is this true? And can somebody give a (sketch of a) proof or point me in ...
1
vote
1answer
62 views

Homotopy extension property and contractibility

Definition. A pair $(X,A)$ of topological space $X$ and its subspace $A$ satisfies Borsuk property if for any topological space $Y$ and for any continuous map $f \colon X \to Y$ any homotopy $F_A ...
7
votes
2answers
146 views

How to check if polylines can be untangled?

In a program I'm writing I need to be able to check whether a straight line between two points is homotopic to a polyline between them. For example in the below example the first one is equivalent to ...
1
vote
1answer
43 views

Left and right homotopies have the same equivalence relations

Let $f, g \colon A \longrightarrow X$ be continuous functions of topological spaces. We say that $f,g$ are left-homotopic if there exists a continuous map $H \colon A \times I \longrightarrow X$ such ...
5
votes
2answers
93 views

Path Connectedness in Van Kampen Theorem

On page 17 of this pdf, http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, the Van Kampen Theorem is proven. That is it is shown that for any covering of a space $X$ by a family of open ...
4
votes
0answers
56 views

$J$-homomorphism and homotopy

We have Bott periodicity theorem for unitary group $U(n)$: $$ \pi_{i-1}^{s}(U) = \pi_{i-1}(U(m)) \simeq \pi_{i}(Gr_m(\mathbb{C}^{2m})) \simeq \pi_{i+1}(SU(2m)) \simeq \pi_{i+1}^{s}(U) .$$ So we can ...
4
votes
2answers
131 views

double comb space is not contractible

I'm trying to show that the double comb space is not contractible. Intuitively I can see why this is true, but I can't seem to formalize a prof. I try to do the following: Let $D$ be the double ...
0
votes
1answer
46 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
6
votes
1answer
106 views

For a transitive action of a path-connected group, does every path lift?

Let $G$ be a Hausdorff, path-connected group acting transitively on a Hausdorff space $X$. Assume the action is continous (i.e. $(g,x) \mapsto g \cdot x$ is continuous) and transitive. It follows ...