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Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

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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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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 ...
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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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50 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
33 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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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. ...
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32 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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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
66 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 ...
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3answers
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How the two non null-homotopic equivalence classes generate the null-homotopic loop on the torus

I am new in Alebraic Topology. Given the torus, we say that the fundamental group of the torus is generated by two loops (or more exactly two equivalent classes of loops). One writes $\pi=\mathbb{Z}\...
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1answer
73 views

Fundamental Group Contains Infinite Cyclic Subgroup

Let $X \subseteq \Bbb{R}^2$ s.t. $\Bbb{R}^2$ such that $S^1 \subseteq X \subseteq \Bbb{R}^2 - \{(0,0)\}$. Prove that contains $\pi_1(X,(0,1))$ an infinite cyclic subgroup. Here's what I did. Let $j : ...
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2answers
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Geometric presentation of fundamental group of a surface

Let $S = S_g$ be a closed surface. An author of a paper writes: We say $\langle a_1, b_1, \cdots, a_{2g}, b_{2g} \ | \ R \rangle$ is a geometric presentation of the fundamental group $\pi_1(S)$ ...
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1answer
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Fundamental groups of the configuration spaces of all triangles and right triangles

This is a question from a past comprehensive exam: Consider triangles in the plane, with vertices given by non-colinear points as usual. The space $T$ of all plane triangles can be given a natural ...
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Fundamental group of two linked spheres

I want to calculate the fundamental group of this space: $$X=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2-100)[(x-10)^2+y^2+z^2-1]=0\}.$$ I’m not good with this kind of exercises: I know the Seifert-van ...
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Is $\left<a,b,c|aba^{-1}b^{-1},aca^{-1}c^{-1}\right>$ the fundamental group of two tori joined by a circle?

If we connect $S^{1}_a\times S^1_b$ to $S^1_c\times S^1_d$ by identifying $S^1_a\times x_0$ with $S^{1}_c\times x_1$ to form $T'$ is $\pi_1(T')=\left<a,b,c|aba^{-1}b^{-1},aca^{-1}c^{-1}\right>$. ...
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First fundamental group $\pi_1(X)$ of $\mathbb{R^3} \setminus \{$a circumference $\cup$ a line tangent to a point on the circumference$\}$

I have to find the fundamental group $\pi_1(X)$ of $X=\mathbb{R^3} \setminus \{$a line $\cup$ a circumference$\}$. Line passing through the centre of the circle, we have: $\pi_1(X) \cong\left(\mathbb{...
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1answer
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What is wrong with this application of van Kampen's Theorem?

My understanding of van Kampen's Theorem (simplified to just two neighbourhoods): Let $X$ be a topological space and let $\{N_a, N_b\}$ be a cover of $X$ such that $N_a \cap N_b$ is path-connected (...
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The fundamental group of a topological space is isomorphic with its connected component fundamental group

can you help me with this problem of fundamental groups? suppose that $X$ is a topological space, let's fix a point on $X$ like $p\in{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ ...
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Showing $\langle a,b\mid abab^{-1}\rangle$ and $ \langle c,d \mid c^2d^2\rangle$ are isomorphic.

I computed the fundamental group of the Klein bottle in two different ways and obtained two seemingly different answers: $$ \langle a,b \mid abab^{-1}\rangle $$ and $$ \langle c,d \mid c^2d^2\rangle. $...
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Fundamental group of $\mathbb{R}^3\setminus$ line+2 circles

I am trying to find the fundamental group of the space $X=\mathbb{R}^3\setminus S$ where $S$ is the union of the $z$-axis, and two circles of radius 1 on the $xy$-plane, centered at $(0,0,1)$ and $(0,...
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1answer
74 views

Lemma 81.1 in Munkres

In the proof of Munkres, which is given in the picture below, I don't understand why we need to show that $h,\ h(e_0)=e_1$ exists iff $[\alpha]\in N(H_0)$. So I tried to organize the proof in the way ...
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Classification of double coverings and $H^1(X ; \mathbb{Z}/2\mathbb{Z})$

It is a well known fact (see for example Hatcher's "Algebraic Topology", chapter $1$) that there is a bijection between the $n$-sheeted coverings of $X$ up to isomorphism of covering spaces and the ...
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1answer
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What does the following manifold look like?

In hyperbolic geometry, we know that any complete hyperbolic surface is the quotient $\mathbb{H}^{2}/\Gamma$ where $\Gamma < \text{Isom}(\mathbb{H}^{2})$ a discrete subgroup. We know $\text{Isom}^...
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1answer
40 views

Cannonical group isomorpshim for fundamental group in a path connected space.

My question is related to this one which tells us that fundamental group $\pi_{1}(X,x_0)$ is abelian $\textit{iff}$ for every pair $\alpha$ and $\beta$ of paths from $x_0$ to $x_1$, we have the same ...
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1answer
72 views

Is there a general way to calculate the fundamental group of a quotient space?

Suppose $X$ is a path-connected topological space, and $A$ is a path-connected subset of $X$. My question is, is there a way to calculate the fundamental group of the quotient space $X / A$ in terms ...
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the fundamental group of $X$ is the symmetric group $S_3$, then whether it has a universal cover?

Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover. Try: Based on Hatcher, $X$ has a ...
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Find the fundamental group of $\Bbb C^2 \setminus \{(x,y):xy=0 \}$. [closed]

What is the fundamental group of $\Bbb C^2 \setminus \{(x,y):xy=0 \}$? How do I proceed? Please help me in this regard. Thank you very much.
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Finding fundamental groups of two spaces.

What are the fundamental groups of $(1)$ $\Bbb R^3 \setminus \{\rm { circle} \}$. $(2)$ $\Bbb R^3 \setminus \{\rm {two\ disjoint\ circles} \}$. Please help me in this regard. Thank you very much.
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Finding fundamental groups.

What are the fundamental groups of $\Bbb R^3 \setminus {\rm (two\ parallel\ lines)}$ and $\Bbb R^3 \setminus {\rm (two\ intersecting\ lines)}$. How do I compute those groups? Thank you very much.
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Group of covering Transformation corresponding to proper discontinuous action of a group on a connected space is the group itself.

Suppose $G$ acts properly discontinuously on a connected space $X$. Show that the group $G(X,p,X/G)$ of the covering transformation of $p:X \rightarrow X/G$ is $G$. I have tried in this manner- ...
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105 views

Product of paths is independent of choices

Suppose the topological space $X$ is the union of two open pathwise connected subspaces $U_1$ and $U_2$ whose intersection is also pathwise connected. Let $x\in U_1\cap U_2$ and let $f_k:\pi_1(U_k,x)\...
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1answer
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Uniqueness In Proof For Fundamental Group Of $S^1$

Perhaps a dumb question, but in the standard proof that $\pi_1(S^1) \cong \mathbb{Z}$ (such as the one given in Hatcher), we prove that every loop $\gamma$ in $S^1$ is homotopic to some loop in $S^1$ ...
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Fundamental group of $T/\mathbb Z_2\setminus\{\text{singular points}\}$

Let $T=S^1\times S^1$. There is a $\mathbb Z_2$-action on $T$ defined by $x\sim -x$ (considering $T$ as a quotient of $\mathbb R^2$). The quotient $X:=T/\mathbb Z_2$this has four singular points, say ...
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Prove that doubly punctured theta space is contractible.

This is a different approach to what was suggested by Joel Pereira in my question Fundamental group of theta-space and the doubly punctured theta space in Munkres Topology Example 70.1 Munkres ...
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Is theta space with a hole in upper arc contractible?

Munkres Topology Example 70.1 Let $\theta$ be theta-space, $\theta_a := \theta \setminus \{a\}$ and $\theta_b := \theta \setminus \{b\}$. Let $\theta_{ab} := \theta_a \cap \theta_b = \theta \setminus ...
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1answer
30 views

In theta space, is the lower arc and line segment a deformation retract of punctured theta?

Munkres Topology Example 70.1 Let X be theta-space, U = $X \setminus \{a\}$ and V = $X \setminus \{b\}$. Let $U \cap V = X \setminus \{a,b\}$ be doubly punctured theta-space where $a,b$ are interior ...
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Topological space with binary operation has abelian fundamental group

I have been given this problem: Let $\textit{X}$ be a topological space and $\mu : \textit{X} \times \textit{X} \rightarrow \textit{X}$ a binary operation. Show that if $\mu$ is continuous and $\...
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Fundamental group of theta-space and the doubly punctured theta space in Munkres Topology Example 70.1

Munkres Topology Example 70.1 Let X be theta-space, U = $X \setminus \{a\}$ and V = $X \setminus \{b\}$. Let $U \cap V = X \setminus \{a,b\}$ be doubly punctured theta-space where $a,b$ are interior ...
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Prove “Contractible implies simply connected” using tools in Munkres Topology. Context is theta-space.

I've read this online, but I haven't seen this proved in Munkres Topology. Has it been? If so, where? In any case, here is my attempt to show it using the tools given in the book. Please verify. The ...
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Let $\varphi: G \to G'$ be an injective homomorphism. If $G'$ is Abelian, then $G$ is Abelian? The context is fundamental groups.

It is known that the image of a homomorphism of an abelian group is abelian (Here is a proof). I have a related conjecture. Let $\varphi: G \to G'$ be an injective homomorphism. If $G'$ is Abelian, ...
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How to find the end point of a lifted path

Let $p: \tilde{X} \to X$ be a covering projection and $x_0 \in X$ be a given point. For a loop $\gamma$ in $X$ based at $x_0$, define the right-action of $\pi_1(X, x_0)$ on the fiber $p^{-1}(x_0)$ as ...
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Prove that there is a bijective correspondence between $π_1(X,x_0)$ and $π_1(X;x_0 → x_1 )$

Let $π_1(X; x_0 → x_1)$ be the set of homotopy classes of paths from $x_0$ to $x_1$ (preserving end points). Prove that there is a bijective correspondence between $π_1(X,x_0)$ and $π_1(X;x_0 → x_1 )$ ...
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Example for a covering map with not equal groups from different base points

I am looking for the examples of a covering map $p: E \to B$ that satisfies following: Let $e_0$, $e_1$ be points of $p^{-1}(b_0)$ and $H_i=p_*(\pi_1(E,e_i))$ Then, $H_0$ and $H_1$ are not eqaul. I ...
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Fundamental group of the union of subspaces

Let $X$ be a topological space, and let $Y$ be a subspace that is a union of nested open subspaces of $X$: $Y=\bigcup Y_n$, where $Y_0\subset Y_1\subset\dots$. Suppose the smallest subspace (hence all ...
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What are some other topological invariants apart from connectedness, compactness and fundamental groups?

I realized that, for any pair of non-homeomorphic topological spaces that I know of, those three invariants are usually sufficient to prove that the two spaces are not the same. So, for example: the ...
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61 views

Fundamental group of the union of a quadric and line

I'm trying to do the following exercise Let be $L$ a line and $Q$ a non-singular quadric in $\mathbb{CP}^2$. How could be $\pi_1(L \cup Q)$? My idea is: $L \cong Q \cong \mathbb{S}^2$ and they can ...
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If $h:S^1\to X$ is not nulhomotopic, then $h_*:\pi_1(S^1)\to\pi_1(X)$ is not trivial.

Munkres' has a proof of the contrapositive of this claim, which seems a little complicated. I was wondering if a simple proof of this statement exists. I think I found one, and was wondering if I ...
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58 views

Fundamental group of $K\# K/\{3\text{ points}\}$

The fundamental group of Klein bottle has been well discussed in many materials, I'm think if it is requested to calculate the fundamental group of $K\# K/\{3\text{ points}\}$ or even more generally $...
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17 views

Fundamental group of a plane united with a circle in 3D space

Let $X=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=9\}\cup\{(x,y,0)\in\mathbb{R}^3|(x-4)^2+y^2=1\}\subset\mathbb{R}^3$ Find the fundamental group $\pi_1(X,p_0)$ where $p_0=(3,0,0)$ Possible solution: I'm ...
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1answer
88 views

The fundamental group of Klein bottle/{3pt}

The fundamental group of Klein bottle has been well discussed in many materials, I'm think if it is requested to calculate the fundamental group of Klein bottle/{3pt} or even more generally Klein ...