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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26 views

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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56 views

Removing one point does not change fundamental group [duplicate]

Let $X$ be a path-connected topological space, $x_0\in X$, s.t. $x_0$ has an open neighbourhood homeomorphic to $\mathbb{R}^d$ for some $d\ge 3$. The task is to show that $\pi_1(X)\cong \pi(X\setminus\...
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1answer
70 views

Why is the fundamental group not a topological group?

It is widely known that group operations in $\Omega X$ are generally not consistent with the natural topology (see, for example, https://arxiv.org/pdf/1105.6363.pdf). Moreover, if the operations of ...
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16 views

Show that if an edge e is on a cycle in the coherent Graph G, then it has to be an edge in a fundamental cycle in T, the spanning tree of G

Definition fundamental cycle: If $T$ is a spanning tree of a given graph $G$, and $e$ is an edge that does not belong to $T$, then the fundamental cycle $C_{e}$ defined by $e$ is the simple cycle ...
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$\pi_1$-equivalence of CW-complexes

Definition. We call $\pi_1$-equivalence the closure of a binary relation between topological spaces "there is a continuous mapping inducing an isomorphism of fundamental groups" to an ...
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27 views

Prove the theorem : $𝜋_𝑛(𝐵)≃𝜋_𝑛(𝐸)⊕𝜋_{𝑛−1}(𝐹)$ [duplicate]

Theorem: For a fiber bundle 𝐹→𝐸→𝐵 such that the inclusion 𝐹→𝐸 is homotopic to a constant map the long exact sequence in homotopy breaks up into split short exact sequences $𝜋_𝑛(𝐵)≃𝜋_𝑛(𝐸)⊕𝜋...
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66 views

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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115 views

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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72 views

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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1answer
68 views

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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2answers
130 views

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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1answer
84 views

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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30 views

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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2answers
93 views

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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34 views

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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2answers
73 views

How do I compute the fundamental group of a disk?

I need to show that $\Pi_1(D^2,x)=\{id\}$ for all $x$ in the Disk. I somehow don't see how I can compute the fundamental group, since we only have the following definition: The fundamental group of $...
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1answer
90 views

Find the fundamental group of a topological space X obtained from S3 by removing a figure 8

Find the fundamental group of $S^3$ = {$(x, y, z, w)$ $\in$ $\mathbb{R}^4$ : $x^2 + y^2 + z^2 + w^2 = 1$} by removing from it a figure eight, that is, the set: $K$ = {$(x, y, $$\frac{1}{\sqrt2}$,$0)$ $...
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1answer
38 views

Seifert-Van Kampen $S^1 \vee S^1$

I would like to use the fact that if two path-connected pointed topological spaces $(X,p)$ and $(Y,q)$ admit two contractible open neighbourhoods of $p$ and $q$, then $$ \pi_1(X\vee Y) = \pi_1(X)*\...
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1answer
37 views

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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1answer
47 views

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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1answer
69 views

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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40 views

Could $nT$ be covered by $mT$?

I understand that the torus of genus one, $T$, can be covered by three different spaces, each corresponding to a different subgroup of $\pi_1 (T) = \mathbb{Z} \times \mathbb{Z}$: the plane $\mathbb{R}^...
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1answer
41 views

Interpretation of fundamental group

I learned the concept of the fundamental group in Munkres where For a space $X$ and $x\in X$, the fundamental group is defined by $\pi_1(X,x)=\Omega(X,x)/\sim$ where $\Omega(X,x)=\{\gamma:[0,1]\to X:\...
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1answer
58 views

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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46 views

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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42 views

Prove that the fundamental group functor preserves products

Is my proof correct? Theorem: Let $(X, p)$ and $(Y,q)$ be pointed topological spaces. Then we have that $$\pi_1(X\times Y, (p,q))=\pi_1(X,p)\times\pi_1(Y,q).$$ Proof: Let $I=[0,1]$ and $\partial I=\{0,...
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42 views

Prove that there exists a homeomorphism between covering spaces

Prove that if two covering maps $f:M{\to}N$ and $f^{\prime}:M^{\prime}{\to}N$, with the same base manifold $N$, are such that the subgroups $f_{\ast}{\pi}_{1}(M)$ and $f^{\prime}_{\ast}{\pi}_{1}(M^\...
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1answer
50 views

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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1answer
72 views

Why is the fundamental group of the Volodin space $X(R)$ the Steinberg group $St(R)$?

The Volodin space $X(R)$ is defined in A.A. Suslin's "On the Equivalence of K-Theories" (https://www.tandfonline.com/doi/abs/10.1080/00927878108822666) as the union of classifying spaces $\...
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1answer
53 views

Fundamental group determines fundamental groupoid?

If $X$ and $Y$ are path-connected topological spaces such that the fundamental group of $X$ is isomorphic to the fundamental group of $Y$, does it follow that the fundamental groupoid of $X$ is ...
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1answer
25 views

Interpreting a topological group with fundamental group $\mathbb{Z}$

We can interpret a topological group with fundamental group $\mathbb{Z}_2$ as - if we traverse any loop on this group twice, we return to the same state. Similarly, for fundamental group $\mathbb{Z}_n$...
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2answers
103 views

Infinite and non-abelian fundamental group

$\require{AMScd}$ I ran into trouble while trying to answer this question. I am trying to prove the following: Suppose $U_1,U_2$ and $U_3 := U_1 \cap U_2$ are open, path-connected subsets of $X = U_1 ...
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67 views

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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91 views

Is the functor $\pi_{1}$ essentially surjective?

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)...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
4
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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? ...
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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+...
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1answer
46 views

Simplicial complex with prescribed fundamental group

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.
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27 views

Elementary proof that the Griffiths twin cone is not simply connected

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\...

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