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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Explicit description of $\pi_1(M)$-action on the relative homotopy groups $\pi_k(\Omega_0M,M)$

Let $M$ be a path-connected space (say, a connected manifold) and denote by $\Omega_0 M$ the based loop space component of the constant loop $x_0$. If $c \colon M \to \Omega_0 M$ denotes the obvious ...
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If the universal cover of $X$ is contractible, then $\pi_n(X) = {0}$ [duplicate]

I know that a covering $p:X\to Y$ is universal if the space $X$ is simply connected. But I'm not sure how do I prove that If the universal cover of X is contractible, then $\pi_n(X) = {0}$ for $n>1$...
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Fundamental group of $\mathbb{R^\times}/\sim$

This is not a homework problem. I was curious about the fundamental group of $X:=\mathbb{R}^\times /\sim$, where $\sim$ is generated by the relation $x \sim x^2+1$. My attempts where to try to ...
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Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

First, let me state some basics mainly coming from Introduction to foliations and Lie groupoids written by I. Moerdijk and J. Mrcun. A codimension $q$ foliation $\mathcal{F}$ on a smooth n-manifold $M$...
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Problem handling free groups in algebraic topology

Trying to compute the fundamental group of a topological space $X$ I have come to the equality $$\pi_{1}(X)\cong\frac{\ast_{i=1}^{n}\mathbb{Z}}{G}$$ where $\ast$ means taking the free product ($n$ ...
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Fundamental Group of the Long Line

Let $\omega_1$ be the first uncountable ordinal. The long line $L$ is defined as the cartesian product of the first uncountable ordinal $\omega _{1}$ with the half-open interval $[0,1)$ equipped ...
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Same homology but different fundamental group

Let $M$ and $N$ be connected, closed, orientable $3$-manifolds. Then I'm able to prove the following statement : If $\pi_1(M)\simeq\pi_1(N)$ then $H_i(M)\simeq H_i(N)$ for all $i$. But I wonder if ...
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Fundamental group of cube with interiors of edges removed?

I'm trying to understand why my calculation of this fundamental group doesn't work: In my diagram, the first figure is a cube, with the sides identified in the obvious way. Via homotopy equivalence, ...
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Compute the fundamental group of $\mathbb{R}^4\backslash \mathbb{R}^2$ [closed]

How would I compute $\pi_1(\mathbb{R}^4\backslash \mathbb{R}^2)$? I'm having trouble writing down an explicit homeomorphism or deformation retraction between $\mathbb{R}^4\backslash \mathbb{R}^2$ and ...
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Surfaces in relation to basic topology.

I need to calculate the fundamental group of the space obtained by punching k holes in the sphere. I understand the spaces H(p) and M(q), with H(p) being the sphere with p handles added and M(q) as ...
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fundamental group of SO(3) through action

I want to show that the fundamental group of SO(3) is $\mathbb{Z}/2\mathbb{Z}$ using the following theorem. if $X$ is locally path connected and simply connected and $G$ is groups with a properly ...
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If higher homotopy groups are trivial, then the fundamental group is a complete invariant?

Let $X$ and $Y$ be two n-manifolds all of whose higher homotopy groups are trivial, and the first homotopy groups are isomorphic (but the existence of a mapping inducing an isomorphism is not assumed)....
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Finding $n$-sheeted path-connected covering spaces of the wedge sum $X\vee Y$ of two spaces.

All references/definitions are from Hatcher. Suppose I have two path-connected, locally path-connected, and semilocally simply-connected spaces $X$ and $Y$ and I want to enumerate the $n$-sheeted path-...
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Normal closure in free group

Let $G=\pi_1(\mathbb{C}-\{z_1,z_2\})$ be the fundamental group of the plane punctured twice. There exist two homotopy classes of loops $\mathfrak{g}_1=[\gamma_1]$ and $\mathfrak{g}_2=[\gamma_2]$ such ...
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Fundamental Group of Polygons

In Kosniowski's book "A First Course in Algebraic Topology", he used Van Kampen's theorem to calculate the fundamental group of given pictures. He defined the first open set, for example for ...
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Simple closed curve in $\mathbb{R}^2\backslash \{x_1,...,x_n\}$ as product of generators of fundamental group

Consider $U=\mathbb{R}^2\backslash \{x_1,...,x_n\}$, by Seifert-Van Kampen theorem $\pi_1(U,p)\cong \underbrace{\mathbb{Z}*\mathbb{Z}*...*\mathbb{Z}}_{n-times} \ \forall \ p\in U$ and we can choose as ...
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Fundamental Group of $\mathbb{R}^2-(\mathbb{Z} \times \{0\})$

In Fundamental group of $\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\})$ it is said because $Y=\mathbb{R}^2-(\mathbb{Z}\times \{0\})$ has a countable points less than whole $\mathbb{R}^2$, then at ...
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We know that for each topological space (say path connected), $\pi_{1}(X)$ is a group, the fundamental group of $X$. But if we take a group $G$, exists there a topological space X such that $\pi_{1}(X)... 0answers 15 views Projective representation of restricted indefinite orthogonal group$SO^+(p,q)$For the restricted Lorentz group$SO^+(1,3)$, all of its projective representations are of the form \begin{equation} D(\Lambda_1)D(\Lambda_2) = \pm D(\Lambda_1\Lambda_2) \end{equation} As far as I ... 0answers 38 views Nonseparating curve on a surface and HNN extension Let S be a closed surface and$\gamma$be a nonseparating simple closed curve. I was told that$\pi_1(S)$can be written down as an HNN extension of$\pi_1(S\setminus\gamma)$. Why is that true? What ... 0answers 43 views Examples of Nori fundamental groups I'm trying to learn the theory of Nori's fundamental group and find myself lacking some good examples to have in mind. The two examples I've seen are the following: The Nori fundamental group of ... 0answers 128 views What were Poincaré's fundamental groups? (a motivation request) In this math.overflow question: https://mathoverflow.net/questions/143116/where-does-the-notation-pi-1x-x-for-the-fundamental-group-first-appear, an answer posits that Poincaré regarded the ... 1answer 46 views Induced homomorphisms and extendibility of continuous functions I recently happened to come across this interesting result in topology: Let$A$be a subspace of$\mathbb{R}^n$and let$h:(A,a_0) \rightarrow (Y,y_0)$. Then, if$h$is extendable to a continuous map ... 0answers 27 views construction a smooth geometrically connected curve via a open subgroup of arithemetic fundament group Let$k$be a field,$U$be a smooth, geometrically connected curve over$k$,$K$is function filed of$U$,$\pi_{1}(U,u)$is etale fundament group,$u$is a suitable geometric point, Let$H$be open ... 0answers 32 views Is an homotopy class represented by the homotopy class of inclusion as subspace? The following question is related to this other question. In the question I have to show the exactness of$$\pi_1(A,x_0) \overset{i_*}{\longmapsto} \pi_1(X,x_0)\overset{j_*}{\longmapsto} \pi_1(X,A,x_0)... 2answers 83 views If$A$is circle then why is$A $contractible? Show that there are no retractions$r: X \rightarrow A$in the following cases: (c)$X = S^1 \times D^2$and$A$the circle shown in the figure. My attempt : I found the answer here But i didn't ... 1answer 66 views the proof of lemma 55.3 in Munkres's book Topology (second edition) What does the fact that$[p_0]$generates the fundamental group of$S^1$mean? I don't know where this fact was used. A part of Lemma 55.3 in the book proves the following: Let$h:S^1\to X$be a ... 2answers 102 views Detect a knot from its fundamental group I'm studying the braid closure and I ended up with the knot$K= (\sigma_1\,\sigma_2\,\sigma_1\,\sigma_2\,\sigma_1)_*$, here the notation is according Murasugi but it does not really matter. By using a ... 0answers 72 views Generators of$\pi_1(\mathbb{RP}^n)$There's a proof in Tom Dieck p.$436$of the Borsuk-Ulam Theorem which is the following: What I don't get here is why taking a generic path$v$from$x$to$-x$we know that composed with the orbit ... 1answer 115 views why they have the same first coordinate in torus? Note: It humble request to everyone please don't downvote my post .Im not genius or intelligent.Im new to the Algebraic Topology.Try to understand my level of understanding.I feel discourage ... 1answer 68 views Fundamental group of a compact surface Let$\mathbb{X}$be a compact surface (possibly with nonempty boundary). If$\pi_1(\mathbb{X})$has an element of order$2$, can we prove that$\mathbb{X}$is homeomorphic to the projective plane? ... 1answer 61 views Fundamental group of$n$-punctured torus I think that very similar questions have been asked elsewhere on MSE, so if there is nothing new here I apologize. I know that the fundamental group of the$n$-punctured torus is the free group$F_{n+...
Let $G$ be a finitely presented group. Can we find a finite simplicial complex $X$ such that $\pi_1(X)=G$ and $\pi_2(X)=0$? I know the conclusion is true if we only require $X$ to be a CW-complex.
The Hawaiian earring $\mathbf{H}$ is the union of the circles $C_n \subseteq \mathbb{R}^2$ with radius $\frac{1}{n}$ and centre $(\frac{1}{n}, 0)$. The Griffiths twin cone is then \$\mathbf{TC} := C\...