# Tagged Questions

For questions about or involving the fundamental group.

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### Computing the fundamental group

I want to compute the fundamental group of the double Torus using the Seifert-van Kampen theorem so then I choose $U=\text{double Torus} / \{\text{point} =x_1\}$ and $V=D$ the disc. The thing is ...
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### Unique homomorphisms between fundamental groups of topological spaces

Let $u:A\rightarrow B$ be a continuous map of topological spaces, $a\in A$, $b=u(a)$. How do I prove that there exists a unique group homomorphism $$u':\pi_1(A,a)\rightarrow\pi_1(B,b)$$ ...
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### Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology.

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology. I want to show this by showing that there does not exist any path $f$, $\forall x, y \in X$. Assume for ...
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### Does inclusion give an injective homeomorphism $\pi[G \cap B(a, r),a] \to \pi[G, a]$?

Let $G$ be a open, path connected subset of the plane and $a\in G$. Set $G' = G\cap B(a,r)$, where $B(a,r)$ is the open ball of center $a$ and radius $r$. Let $\gamma$ be a closed loop in $G'$ with ...
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### When does triviality of $f_*$ imply null-homotopy of $f$?

I've been considering problems of the following type: Given certain topological spaces $X$ and $Y$ and a continuous function $f:X\to Y$, prove that $f$ is null-homotopic. The cases studied so ...
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### Simply connected means universal-covering spaces

A covering space $p:(Y,y_0) \to (X,x_0)$ is called universal when $Y$ is simply connected (say that we restrict ourselves to path connected spaces, and locally path connected). I heard that this ...