# Tagged Questions

1answer
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### need help with problem on homology group

Let $A_n=\{z\in \mathbb{C}\mid z^n$ is non-negative real number$\}$ then find $H_1(A_n,A_n-\{0\})$ $H_1(A_n,A_n-\{z\})$ when $0\not=z\in A_n$ show that $A_n$ is not homeomorphic to $A_m$ when ...
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### Prove these 3 spaces are homotopy equivalent

The image is below. (a) $S^2$ with a diameter. (b) $T^2$ with a disk in the middle hole. (c) $S^2$ tangent with $S^1$ . I think they may the deformation retract of the same space. But I can't ...
1answer
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### How to show that: A path is homotopic to a given point, then it is homotopic to any other point.

Let $D \subset \mathbb C$ be a domain and $\gamma : [\alpha, \beta] \to D$ a closed path. Let $a \in D$ be a given point. Assume that $\gamma$ is homotopic to that point $a$. Prove that $\gamma$ is ...
0answers
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### Prove that $S_R^+(b)$ and $S_r^+(a)$ are homotopic in a domain $D$.

currently I'm working on the following exercise: Let $D \subset \mathbb C$ be a domain. Let $a,b \in \mathbb C$ and $r,R > 0$ such that $B_r(a) \subset B_R(b)$ and \begin{align*} A := \{z ...
1answer
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1answer
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2answers
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### When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
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1answer
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### Homotopy equivalence of two spaces, homework

I need to prove that spaces $\mathbb R^2 \setminus \{e_1,-e_1\}$ and $S(e_1,1)\cup S(-e_1,1)$ are homotopy equivalent. $e_1$ is basis vector $(1,0)$ and $S(e_1,1)$ is a sphere centered on $e_1$ with ...
0answers
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### The inclusion into the mapping cocylinder need not be a cofibration

The mapping cocylinder of a map $f:X\rightarrow Y$ is given by $N_f=\{(x,\beta)|f(x)=\beta(0)\}\subseteq X\times Y^I$. $f$ factors through a homotopy equivalence $j:X\rightarrow N_f$ given by ...
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### homotopy groups of mapping space

I got this homework problem: $X,Y$ finite CW-complexes with $\dim X=m$ and $Y$ is $n$-connected. Prove that $\pi_k(map(X,Y))=0$ for all $k \le n-m$. Thanks for the help!
0answers
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### Contractible and Compact space can be contained in an open set after time $t_0$?

$X$ is a topological space that is contractible and compact. Show that if $U$ is an open set in $X$ containing $x_0$ then there exists $t_0<1$ so that $H(x,t)∈U$, for all $x∈X$, and all $t_0≤t≤1$. ...
1answer
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### homotopy of circles

Consider the circle with center in $(0,0)$ and radius 1 and the circle in $(2,0)$ and radius 1.5 in the plane. Are they homotopic (a) if we remove the origin, (b) if no point is removed? I think that ...
1answer
771 views

### Homotopy equivalence of universal cover

As part of am exam question (Q21F here), I'm trying to prove that if $X$ and $Y$ are path-connected, locally path-connected spaces with universal covers $\widetilde{X}$ and $\widetilde{Y}$, ...