Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

948 questions
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Fundamental group of the complement of $k$ points in $\mathbb{R}^2$

Let $S = \{p_1, \ldots, p_k\}$ be a set of $k$ points in $\mathbb{R}^2$ ($1 \leq k < \infty$). My goal is to calculate the fundamental group of $\mathbb{R}^2 \setminus S$ using van Kampen's theorem....
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If $E \rightarrow X$ is a covering, then $|\pi_1 (X)|$ divides to $\chi (E)$

I want to prove that no finite group acts free in $\mathbb{R}^n$ in the process I found the following doubt: How can you prove that: If $E \rightarrow X$ is a covering, then $|\pi_1 (X)|$ ...
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Bundle over circle; Monodromy action on cohomology of fiber

I want to understand the following sentence: Let $\pi: M \to S^1$ be a fiber bundle with path connected fiber $F$. Then its monodromy action on $H^k(F; \mathbb C)$ satisfies... How is the ...
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Fundamental group of group of homeomorphism of a compact surface

I'm reading "A Primer on Mapping Class Group", and there is something I don't understand in the proof of Theorem 4.6. Define $\mathrm{Homeo}^+(S)$ to be the group of orientation-preserving ...
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Using Van Kampen's Theorem to determine fundamental group

I'm trying to calculate the fundamental group of a surface using (i) deformation retracts and (ii) Van Kampen's Theorem. I'm really struggling understanding the group theory behind it and the ...
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Equivalence between Category of Covers and $\pi_1(X)$ Sets

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 38): In order to show the category equivalence claimed in Thm 2....
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Notions of Fundamental Groups for semisimple algebraic groups

Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which ...
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How to see correspondence between G-covers and homomorphisms

I found this paper on van Kampen's theorem (https://www3.nd.edu/~andyp/notes/SeifertVanKampen.pdf), and I was wondering how to prove Lemma 1: Let Z be a reasonable nonempty path-connected space, let ...
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Algebraic Fundamental Group $π_1(U)$

I have a $1 \frac{1}{2}$questions about two examples introduced in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122): Let $X$ be an integral proper normal $k$-...
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Homotopy equivalence between quotients by free actions

Let $X,Y$ two contractible spaces. Assume there is a free action of a group $G$ on both spaces. $X$ and $Y$ are obviously homotopy equivalent. In particular, we can consider the homotopy equivalence ...
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Real plane with two holes and circumference are not homotopy equivalent

I´d like to obtain an argument to prove that the real plane with two holes, for example $\mathbb{R} \setminus \{p,q\}$ is not homotopy equivalent to the circumference $S^1$. I know they have ...
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Can I use Seifert-van Kampen theorem infinite times

I know the definition of Seifert-van Kampen theorem for a topological space "made" with 2 parts. Is not difficult to see that if I use the theorem a finite number of times to calculate a the ...
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fundamental group of torus minus a point

I want to calculate fundamental group of torus minus a point using Van-Kampen Theorem.I know that result is $\Bbb Z* \Bbb Z$. I proved to consider like open $U$ the torus minus a disk $S$ that ...
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Simply connected and direct products

A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, that is, $\pi(X,x_0) = 0$ for any $x_0 \in X$. Now, we know ...
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Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space? Is ...
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Fundamental group of $\mathbb{R}P^2$ in 2 models

I know that $\pi_1(\mathbb{R}P^2)\cong\mathbb{Z}_2$, but in the square model, I get that $\pi_1(\mathbb{R}P^2)=\langle a,b\colon abab\rangle$. These groups must be isomorphic, but I can't find the ...
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Fundamental group of $\mathbb{C^*}/G$

Calculate the fundamental group of $\mathbb{C^*/G}$ where $G=\{\phi^n:n\in \mathbb{Z}\}$ is the group of the homeomorphisms s.t. $\phi (z)=2z$ and $\mathbb{C^*}=\mathbb{C}-\{0\}$ Any hints on how to ...
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Using the Hatcher's notation on proposition 1.39. Let $G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$ be the Deck transformation group for the covering space $p: \tilde{... 1answer 22 views Extension of surface group by cyclic is residually finite Let$G$be the fundamental group of a surface, and consider an extension$1 \to \mathbb{Z}/p\mathbb{Z} \to E \to G \to 1$. Is$E$residually finite? I'm interested in proving the injectivity of the ... 1answer 28 views Same cell structure have isomorphic fundamental groups Let$X$and$Y$be two topological spaces both with the following cell structure 2 0-cells 4 1-cells 3 2-cells 1 3-cell Can I conclude anything about the fundamental groups of$X$and$Y$? Are ... 1answer 63 views Fundamental group of tetrahedron Consider the following double tetrahedron We glue$DCE$to$CBA$,$CBE$to$BDA$and$BDE$to$DCA$. We call the resulting space$L$. I want to find a cell-structure on$L$with only two$0$-cells ... 0answers 83 views Modify the Alexander horned sphere for an embedding$S^2 \hookrightarrow ֓\mathbb{R}^3$s.t. neither component of$\mathbb{R}^3 − S^2$is 1-connected. Question 2.B.6 in Allen Hatcher's Algebraic Topology page 176: Modify the construction of the Alexander horned sphere to produce an embedding$S^2 \hookrightarrow ֓\mathbb{R}^3$for which neither ... 0answers 60 views Computing fundamental group of a circle I'm very new to Topology but we just covered in class an algorithm for calculating the Fundamental Group of a surface given a triangulation. The algorithm goes as follows: 1) Find a spanning tree in ... 1answer 49 views What is the fundamental group of the$3$-manifold bounded by a genus-$2$torus? As in the question, let$X$be the$3$-manifold bounded by$\partial X$which is a torus of genus$2$.What is$\pi_1(X)$? I noted that$\pi_1(\mathbb{R} ^3\backslash X) \cong \langle a, b | aba^{-1} ...
Let $A:\mathbb{C}^*\to\mathrm{M}_n(\mathbb{C})$ be a holomorphic map. Consider the system of first order differential equations \begin{equation} \begin{cases} \frac{dY}{dz} = A Y\\ Y(1)= I, \end{cases}...