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Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

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3answers
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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
37 views

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
18 views

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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0answers
16 views

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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0answers
71 views

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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0answers
34 views

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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0answers
14 views

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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20 views

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
44 views

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
42 views

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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30 views

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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1answer
20 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 ...
2
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1answer
58 views

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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1answer
26 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 ...
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1answer
51 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 ...
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0answers
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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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0answers
45 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 ...
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1answer
39 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} ...
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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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1answer
63 views

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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1answer
47 views

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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2answers
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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
26 views

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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1answer
29 views

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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2answers
75 views

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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2answers
98 views

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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1answer
33 views

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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2answers
78 views

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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1answer
35 views

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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1answer
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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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1answer
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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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1answer
63 views

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
21 views

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
56 views

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, ...
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2answers
36 views

Surjectivity of sending loop from fundamental group to the endpoint of its lift?

This question refers to the following map defined on a fundamental group of some topological space $X$ covered by a covering map $p:\bar{X}\rightarrow X$: $$r: \pi_{1}(X, x_{0}) \rightarrow p^{-1}(x_{...
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1answer
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Fundamental group of uncountable space with cofinite topology

So it is easy to show that such a space is path connected (assuming CH, injective maps $f:I \rightarrow X$ are continuous) but I'm not sure how to start computing the fundamental group. Will it depend ...
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0answers
38 views

Homotopy groups of $\mathbb R^n-\mathbb Z^n (n>1)$?

How to compute the homotopy groups of $\mathbb R^n-\mathbb Z^n (n>1)$? The fundamental group may be an easy exercise using Van Kampen theorem, but how about higher ones?
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3answers
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How is going around the circle once in each direction homotopic to a point?

Two paths are homotopic if one can be continuously deformed to the other, right? So I've been told that the fundamental group of the circle is isomorphic to the integers, since you can't deform e.g., ...
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2answers
54 views

find two topological spaces $Y_1$, $Y_2$ such that $\pi (Y_1)=\pi (Y_2)= (0)$ but $\pi (Y_1 \cup Y_2)\ne (0)$

I everybody, I have to solve this exercise. I have to find two topological spaces $Y_1$, $Y_2$ such that $\pi (Y_1)=\pi (Y_2)= (0)$ but $\pi (Y_1 \cup Y_2)\ne (0)$. Can you help me?
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1answer
108 views

Hatcher Exercise 1.2.8

I am trying to prove the following excersise (1.2.8) from Hatcher's Algebraic Topology: Given 2 tori $S^1\times S^1$ and identifying $S^1\times \{x_0\}$} compute the fundamental group. My approach is ...
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1answer
51 views

The fundamental group of the Lattice - (R x Z) U (Z x R)

I am trying to show that the identity map $ id:S_h \vee S_v \rightarrow S_h \vee S_v$ does not lift to L = $(\mathbb{R} \otimes \mathbb{Z}) \cup (\mathbb{Z} \otimes \mathbb{R}) $ via the covering ...
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0answers
35 views

Does the center of $\pi_1(Y)$ act trivial on $[X,Y]_\star$?

Let $X$ and $Y$ be based (and well-pointed) and connected. We have an action of $\pi_1(Y)$ on the set $[X,Y]_\star$ of based homotopy classes of based maps. The quotient is just the set $[X,Y]$ of ...
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0answers
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Degree of universal cover of simple Lie group

I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie ...
3
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1answer
65 views

Fundamental group of the $m$-fold suspension of a finite discrete space

Recall that the suspension of a topological space $X$ is the space $SX$ resulting by identifying $X\times\{0\}$ and $X\times\{1\}$ to single points of the "cylinder" $X\times[0,1]$. Now let $X_m$ be a ...
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1answer
53 views

covering spaces and equivalency of these three propositions [closed]

This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following ...
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1answer
35 views

The fundamental group of $(S^1\times S^1)/(S^1\times \{x\})$

What is the fundamental group of $(S^1\times S^1)/((S^1\times\{x\})$ where $x$ is a point in $S^1$? My guess is that $S^1$ is a deformation retract of $(S^1\times S^1)/(S^1\times \{x\})$. Thus $\pi_1(...
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0answers
34 views

Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
2
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1answer
37 views

Fundamental group via deck transformations considering rotation on sphere

Let $Z_m$ act on $S^1$ by multiplication with $e^{2\pi ki/m}$ for $k \in Z_m$. Let $X = S^1 / Z_m$ be the orbit space of this action. Then we have a universal cover $q:S^1 \rightarrow X$ given by the ...
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0answers
54 views

The small étale topos of a scheme is equivalent to the category of finite $\pi_1(X,x)$-sets for every scheme $X$ and every geometric point $x$

Recall Milne, Etale cohomology, Theorem I.5.3: Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/X\to Sets$ ($FEt/X$ being the category of $X$-schemes finite and ...
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1answer
71 views

Homotopy of continuous map from a space with finite fundamental group

Problem source: 2c on the UMD January, 2018 topology qualifying exam, seen here https://www-math.umd.edu/images/pdfs/quals/Topology/Topology-January-2018.pdf I have an argument for it, but I am not at ...