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Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

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Local system associated to monodromy representation

How can I associate a local system to a representation $\rho: \pi_1(X) \to \mathbb C^*$? I have seen some construction, but it doesn't click for me. I know that the idea is to use a diagonal action ...
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1answer
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What is the fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$

I want to know about fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ by Seifert-Van Kampen theorem. In my guessing, that is $\langle a_1, a_2 ,... a_n | a_1^{2}a_2^{2}\cdots a_n^{2}=1\rangle$. ...
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Classification of compact connected manifolds by fundamental group [duplicate]

Every compact connected 2-manifold (I define this as a surface) is homeomorphic to a 2-sphere, a connected sum of tori or a connected sum of projective planes. Since the fundamental groups of the ...
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1answer
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Automorphism covering space and universal cover

I have again a new question about the automorphism of covering space and the universal cover of a topological space $B$. Actually, let $p : X \rightarrow B$ the universal cover of $B$. I take a normal ...
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2answers
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How can I visualise the fundamental group of the projective plane?

The real projective plane $\mathbb{RP}^2$ has fundamental group $C_2$. We can understand this via the universal covering mapping $S^2 \to \mathbb{RP}^2$ which identifies antipodal points: the ...
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1answer
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Let $f : B^2 \rightarrow \mathbb{R}^2$ a continuous map, is $B^2 \subset \operatorname{im}(f)$?

Consider a continuous map $f : B^2 \rightarrow \mathbb{R}^2$ such that $f(S^1) \subset S^1$ and $deg(f_{|S^1}) \ne 0.$ Prove that $B^2 \subset \operatorname{im}(f).$ [Note: Here, $B^2$ is the ...
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Problem with set of cycles covering a graph

My question is the following: is there always, for any non-directed graph $G$, a choice of generators of the fundamental group or, more in general, a set of cycles, covering the graph and with the ...
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Fundamental group of the complement of $3$ disjoint hypersurfaces in $\mathbb{C}P^2$

Let $X$ be the union of $3$ hypersurfaces in $\mathbb{C}P^2$, then how to compute the $\pi_1(\mathbb{C}P^2\setminus X)$? What I know is the complement of a hypersurface in $\mathbb{C}P^2$ is $\mathbb{...
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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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1answer
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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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1answer
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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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1answer
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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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Constructing a Universal Cover--Proving Injectivity

Here is a quote from Hatcher's Algebraic Topology: Given a set $U \in \mathcal{U}$ and a path $\gamma$ in $X$ from $x_0$ to a point in $U$, let $$U_{[\gamma]} = \{[ \gamma \cdot \eta ] \mid \eta \...
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Pencil and fundamental group

Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_{a,b}=a(f_2)^3+b(f_3)^2$, we have a map $\phi:\mathbb{C}P^2\setminus B\to \mathbb{C}P^1$, where $B$ is the base locus of ...
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Homomorphism Induced by Inclusion is Trivial

On page 64 of Hatcher's book on Algebraic Topology, he writes the following: ...if the map $\pi_1(U) \to \pi_1 (X)$ is trivial for one choice of basepoint in $U$, it is trivial for all choices of ...
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Is this proof that $\widehat{G_p}$ is pro-$p$ free correct?

Let $G$ be an abstract group with the following presentation: $$G \simeq \langle x,y \mid x^2y^2 = 1 \rangle $$ Let $p \neq 2$ be an odd prime. I want to show that $\widehat{G_p} \simeq \mathbb{Z}_p$...
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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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1answer
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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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1answer
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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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Deck Transformations Question

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{...
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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 ...
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Annulus in three dimensions [closed]

Let $\delta>0$, and consider $$T=\{\,x\in\mathbb{R}^3\colon \, \,\delta<\|x\|<2\delta\,\}$$ What is a simple argument to show that $T$ is simply connected?
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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} ...
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Monodromies of complex differential equation

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}...
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Constructing the flat vector bundle associated to a given linear representation of the fundamental group

I'm reading this notes, and I have some questions about the contruction on page 18. Let $M$ be a connected manifold and $E\rightarrow M$ a flat vector bundle over $M$. Consider $\{(U_\alpha, \phi_{...
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Fundamental group of torus knot without thickening

The calculation of the fundamental group of a $(m, n)$ torus knot $K$ is usually done using Seifert-Van Kampen theorem, splitting $\mathbb{R}^3\backslash K$ into a open solid torus (with fundamental ...
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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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1answer
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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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Can we always view loops as maps from $S^1\to X$?

I am working on exercises from Hatcher, algebraic topology and in a certain exercise (1.1.5), we identify a loop, which is a path $\gamma:I\to X$ with a map $f:S^1\to X$, which we can do since $\gamma(...
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Find finite path connected topological space with $π_1 (X, x_0 ) \cong \Bbb Z/2\Bbb Z$

Find a finite non-empty topological space $X$ that satisfies: $X$ is path connected Its fundamental group $π_1 (X, x_0 ) \cong \Bbb Z/2\Bbb Z$ for $x_0 \in X$ Is the connected pair $\{\{0\},\{1\},\{...
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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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1answer
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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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1answer
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Let $X$ be a connected CW complex and $G$ a group such that every $\pi_1(X)\to G$ is trivial. Show that every $X\to K(G, 1)$ is nullhomotopic.

Question 2 in Chapter 1.B in Hatcher's Algebraic Topology: Let $X$ be a connected CW complex and $G$ a group such that every homomorphism $\pi_1(X)\to G$ is trivial. Show that every map $X\to K(G, ...