# Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

950 questions
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### Compute fundamental group _visually_ by the polygonal representation of the space

Some time ago, before I learnt about covering spaces and Seifert-Van Kampen theorem, I tried to compute visually the fundamental group of some spaces. For example I figure out by myself that the ...
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### References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$

I have read (Theorem 29.3, line 7, pg 144) the following statement: $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$. and ...
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### Proof by induction of path composition

Let $w, \alpha_n: I=[0,1] \rightarrow \mathbb{S}^1$, $w(s)=e^{2\pi i s}$, $\alpha_n(s)=e^{2 \pi i n s}$. Let $[u] \in \pi_1(\mathbb{S}^1,1)$ be the path homotopy class of the path $u$, an element of ...
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### Fundametal group and continuous extension to $\Bbb R^n$

Let $X ⊂ \Bbb R^n$ be a non empty subset with $n>0$ and let $x_0 ∈ X$. Let $Y$ be a non empty topological space and $g : X → Y$ a continuous map. Suppose $g$ has a continuous extension defined on ...
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### What can we say about two topological spaces with the same fundamental group?

Let's consider two topological spaces. If they are homeomorphic, or homotopic equivalent, they have isomorphic fundamental groups, but the converse is not true. My question is: is there a (non trivial)...
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### Hatcher Lemma 1.19

Let $f$ be some loop about $x_0$. From what I understand, we want to show that $\varphi_{0*}([f]) = \beta_h\varphi_{1*}([f])$ or $[\varphi_{0} f] = [h \ast (\varphi_{1}f) \ast \overline{h}]$ From what ...
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### First fundamental group and connectedness of $X:=\mathbb{R^4}\setminus \{\pi_1 \cup \pi_2 \}$

On $\mathbb{R^4}$ consider $\pi_1 := \{x_1=x_2=0\}$ and $\pi_2 :=\{x_3=x_4=0\}$. Let $X:=\mathbb{R^4}\setminus \{\pi_1 \cup \pi_2 \}$ . Show that $X$ is arc-connected and find $\pi_1 \left(X\right)$...
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### Loops as maps from $S^1\to X$, Hatcher 1.15 [duplicate]

I am working on Hatcher's problem 1.1.5. Show that for a space $X$, the following three conditions are equivalent. $\textit{a)}$ Every map $S^1\to X$ is homotopic to a constant map. $\textit{b})$ ...
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### If $f_0=\omega_m$ and $f_1=\omega_n$, why is it not automatic that $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$?

In the second paragraph of Hatcher's proof, fourth sentence, it says The uniqueness part of (a) implies $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$. How does the uniqueness part ...
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### Finding an open simply connected subset in a punctured open simply connected set

Let $X$ be an open , simply connected (path connected with trivial first homotopy group) subset of $\mathbb R^2$. Let $0\in X$. Is it true that for every $p,q\in X\setminus \{0\}$, there is an open ,...
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### Cannonical group isomorpshim for fundamental group in a path connected space.

My question is related to this one which tells us that fundamental group $\pi_{1}(X,x_0)$ is abelian $\textit{iff}$ for every pair $\alpha$ and $\beta$ of paths from $x_0$ to $x_1$, we have the same ...
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### Is there a general way to calculate the fundamental group of a quotient space?

Suppose $X$ is a path-connected topological space, and $A$ is a path-connected subset of $X$. My question is, is there a way to calculate the fundamental group of the quotient space $X / A$ in terms ...
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### the fundamental group of $X$ is the symmetric group $S_3$, then whether it has a universal cover?

Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover. Try: Based on Hatcher, $X$ has a ...
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### Find the fundamental group of $\Bbb C^2 \setminus \{(x,y):xy=0 \}$. [closed]

What is the fundamental group of $\Bbb C^2 \setminus \{(x,y):xy=0 \}$? How do I proceed? Please help me in this regard. Thank you very much.
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### Finding fundamental groups of two spaces. [closed]

What are the fundamental groups of $\Bbb R^3 \setminus \{\rm { circle} \}$. $\Bbb R^3 \setminus \{\rm {two\ disjoint\ circles} \}$. Please help me in this regard. Thank you very much.
What are the fundamental groups of $\Bbb R^3 \setminus {\rm (two\ parallel\ lines)}$ and $\Bbb R^3 \setminus {\rm (two\ intersecting\ lines)}$. How do I compute those groups? Thank you very much.
Suppose $G$ acts properly discontinuously on a connected space $X$. Show that the group $G(X,p,X/G)$ of the covering transformation of $p:X \rightarrow X/G$ is $G$. I have tried in this manner- ...