# Questions tagged [fundamental-groups]

For questions about or involving fundamental groups of topological spaces, as well as related topics such as fundamental groupoids and étale fundamental groups.

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### some detail in the proof of Invariance of domain theorem in Munkres

the above is from Munkres Topology according to it $f : U\to S^2$ is continuous and injective and $B$ any closed ball contained in $U$ and $a , b$ two points of $S^2-f(B)$.it says because the ...
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### Covering space of compact surface with free fundamental group

Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ? I know that if we ...
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### Hatcher Theorem 1B.8 / Proposition 1B.9 concluding that a map inducing the identity is homotopic to the identity?

In introducing the concept of $K(G, 1)$, Hatcher's Algebraic Topology proves the theorem that the homotopy type of a CW complex $K(G, 1)$ is uniquely determined by $G$, by citing a proposition and ...
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### Homotopy classes of paths on $\mathbb{R}^2 \setminus\{0\}$

Consider the plane with a point removed $M=\mathbb{R}^2\setminus\{0\}$. Let $p_1,p_2\in M$ be two points. I want to understand how to classify the homotopy classes of paths joining $p_1$ and $p_2$. ...
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### fundamental group of a sphere with 2 handles

Prove that the fundamental group of a sphere with 2 handles contains a free group of 2 generators. Intuitively, it is clear that we can cut a figure homotopically equivalent to a bouquet of two ...
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### What exactly is the proof of onto here?

I have been reading the solution of the following question: We can regard $\pi_1(X, x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S^1,s_0) \to (X, x_0).$ Let $[S^1, X]$ be the ...
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### Computing fundamental groups of some figures

As an excercise of my Algebraic topology lessons, I have to compute the fundamental group of the following figures: There is no any information, only the drawing, so I am a little bit lost on how to ...
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### Computing the fundamental group of a quotient space

I have been reading some notes of former students of my Algebraic topology lessons and I have seen this excercise: Let denote $D^2:= \{(x,y) \in \Bbb{R}^2 : x^2 + y^2 \leq 1 \}$ and $S^1=\partial B^2$...
### Prove that there is no function $f: B[0,1] \to \mathbb{S}^1$ such that $\left.f\right |_{\partial B[0, 1]} = id$
I have to prove that there is no function $$f: B[0,1] \to \mathbb{S}^1$$ such that $$\left.f\right |_{\partial B[0, 1]} = id \tag{\star}$$ I want to use the concept of fundamental groups (however ...