# Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

950 questions
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### CW complex with fundamental group $\Bbb Z/n$

I'm just learning about CW complexes. I came across this answer: "One way of constructing a connected $2$-dimensional CW complex with fundamental group $\Bbb Z/n$ (for some integer $n\geq 2)$ is to ...
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### Fundamental group S^2/~

Let $X = \mathbb{S}^2/\sim$ where $(cos(\theta), sin(\theta), 0 ) \sim (cos(\theta + \pi), sin(\theta + \pi), 0 )$ for all $\theta \in [0, 2\pi]$ Calculate fundamental group of $X$ I try use ...
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### Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$

Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$ for $x,y,z$ such that $x^2+y^2+z^2=1$. Calculate the ...
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### Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$. [duplicate]

Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$. I'm trying to solve this problem but I do not know ...
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### Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$. I think that this space is the free product $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$, but I do not know how to ...
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### Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.

Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$. I am trying to use the Van Kampen theorem but I do not know how to apply ...
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### Find a topological space whose fundamental group is $D_4$ [duplicate]

$D_4$ here indicates the dihedral group of order 8. Does there exist any trivial example of topological spaces such that it has $D_4$ as it's fundamental group.
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### computing the fundamental group of a wedge sum

I'm trying to solve the problem $18 (a$ from the chapter $1$ of Hatcher's Algebraic topology. it is: Prove that the wedge sum $S^1 \vee S^2$ has fundamental group $\mathbb{Z}$. The next chapter is ...
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### Induced homomorphism on $S^1$ non-trivial

Let $f:S^1\rightarrow S^1$ be a map such that $f(-x)=-f(x)$ for all $x\in S^1$. Show that the induced homomorphism $f_\ast:\pi_1(S^1,x_0)\rightarrow\pi_1(S^1,f(x_0))$ is non-trivial. What I know: 1. ...
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### Why the only finite fundamental group of compact surfaces are those of $\mathbb{S}^2$ and $\mathbb{RP}^2$?

I know that the fundamental group of orientable surfaces is of the form $F(a_1,b_1,\ldots,a_n,b_n)/N(\prod a_j*b_j*a_j^{-1}*b_j^{-1})$ where $F$ is the free products with certain generators and $N$ is ...
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### Fundamental group of S_3 without curve

Define $S^3$ as $$S^3 = \left\lbrace (z_1, z_2) \mid |z_1|^2 + |z_2|^2 = 2 \right\rbrace$$ and curve $C$ as $$C(t) = (e^{imt}, e^{int})$$ for $m,n \in \mathbb{Z}$, such that gcd(m,n) = 1. The ...
Consider a continuous map $\alpha:A\to B$ with the homotopy lifting propery and the unique path lifting property. Consider the induced fiber functor $$F:\pi_1B\longrightarrow \mathsf{Set}$$ taking a ...
### $X \setminus point$ not path-connected implies $X$ simply connected
Per $X$ be a path-connected space. Does $X \setminus point$ not path-connected implies $X$ simply connected? Thinking about curves and lines seem to suggest the truth of the statement, but I think ...