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.

For a topological space $X$, we can form the set of all loops in $X$ based at a given point $x_0 \in X$, i.e. $\mathcal{F}(X, x_0) = \{\alpha : [0, 1] \to X \mid \alpha\ \text{continuous}, \alpha(0) = \alpha(1) = x_0\}$.

Let $\alpha, \beta$ be two loops in $X$ based at $x_0$; we say $\alpha$ is homotopic to $\beta$ if there exists a continuous map $H : [0, 1]\times [0, 1] \to X$ such that $H(s, 0) = \alpha(s)$ for all $s \in [0, 1]$, $H(s, 1) = \beta(s)$ for all $s \in [0, 1]$, and $H(0, t) = H(1, t) = x_0$ for all $t \in [0, 1]$. We call $H$ a homotopy of paths (or a homotopy relative to $\{0, 1\}$).

Setting $\alpha \sim \beta$ when $\alpha$ is homotopic to $\beta$, we find that $\sim$ is an equivalence relation on $\mathcal{F}(X, x_0)$. We define $\pi_1(X, x_0)$ to be the quotient space, i.e. $\pi_1(X, x_0) = \mathcal{F}(X, x_0)/\sim$. The set $\pi_1(X, x_0)$ obtains a group structure via concatenation of paths. We call $\pi_1(X, x_0)$ the fundamental group of $X$ at $x_0$.

If $x_0, x_1 \in X$ are in the same path-connected component of $X$, then $\pi(X, x_0)$ and $\pi(X, x_1)$ are (non-canonically) isomorphic. In particular, if $X$ is connected, we often supress the base point and just write $\pi_1(X)$ for the fundamental group of $X$.

The fundamental groups at different basepoints $X$ can be assembled into a single object called the fundamental groupoid of $X$ (often written as $\pi_{\leq 1}(X)$ or $\Pi_1(X)$). This is the groupoid whose objects are points of $X$ and in which a morphism $x\to y$ is a homotopy class of paths from $x$ to $y$, with composition given by concatenation of paths. For each $x\in X$, the automorphism group of $x$ as an object of the fundamental groupoid is the fundamental group $\pi_1(X,x)$.

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Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall down. My friend also told me that I need to be…
Anonymous - a group
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$\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable

Question: Show that $\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable. Motivation: This is one of those problems that I saw in Hatcher and felt I should be able to do, but couldn't quite get there. What I Can Do: There are proofs of this…
user2959
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An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I didn't receive an exhaustive answer to this question from my teacher, in…
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Is the fundamental group of every subset of $\mathbb{R}^2$ torsion-free?

It seems that the fundamental group of any subset of $\mathbb{R}^2$ will not have an element of finite order. Though the $3$-dimensional version is an open problem I couldn't immediately see why it is true in the $2$-dimensional case. Please shed…
Dinesh
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The First Homology Group is the Abelianization of the Fundamental Group.

I am trying to understand the proof of the following fact from Hatcher's Algebraic Topology, section 2.A. Theorem. Let $X$ be a path connected space. Then the abelianization of $\pi_1(X, x_0)$ is isomorphic to $H_1(X)$. I am having trouble…
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Every Group is a Fundamental Group

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and for any $x \epsilon X$ there is a fundamental…
Anirban
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fundamental group of the Klein bottle minus a point

I'm trying to see the fundamental group of the Klein bottle minus a point without success. I know how to solve the torus minus a point giving a deformation retraction to the wedge sum of two circles. My solution of the torus minus a point: I need…
user42912
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Calculating fundamental group of the Klein bottle

I want to calculate the Klein bottle. So I did it by Van Kampen Theorem. However, when I'm stuck at this bit. So I remove a point from the Klein bottle to get $\mathbb{Z}\langle a,b\rangle$ where $a$ and $b$ are two loops connected by a point.…
simplicity
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The fundamental group of a product is the product of the fundamental groups of the factors

Hello :) i want to prove the following statement: $\pi_1(X\times Y,(x_0,y_0))\equiv\pi_1(X,x_0)\times\pi_1(Y,y_0)$ But how to do that? Is this just the projection and the use of the product topology? Thank you for help :) I also want to prove that…
Uncountable
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Examples of fundamental groups

I'm starting to study fundamental groups and I didn't find in the books of Algebraic Topology many examples of them. Can you list the examples you know and the demonstrations? I think it would be useful to a self-student and also to a…
user42912
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The fundamental group of the Möbius strip

What is the fundamental group of the Möbius strip? Is it given by $\{-1,1\}$ as the lemma of Synge supposes, or am I wrong and it does not apply there?
AlexisZorbas
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Does there exist a compact submanifold of $\mathbb{R}^3$ whose fundamental group is $\mathbb{Z}^3$?

Does there exist a compact submanifold of $\mathbb{R}^3$ whose fundamental group is $\mathbb{Z}^3$ ? The question in the title is a generalization of the question that really interests me: Does there exist a connected finite set of unit cubes of…
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Deck transformations of universal cover are isomorphic to the fundamental group - explicitly

I have read on several places that given a (say path connected) topological space $X$ and its universal covering $\tilde{X}\stackrel{p}\rightarrow X$, there is an isomorphism $$\mathrm{Deck}(\tilde{X}/X) \simeq \pi_1(X, x_0).$$ Here…
Pavel Čoupek
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Need help on how to compute the fundamental group of a space.

I'm studying for an oral qualifying exam and going through various past exams I find on the interwebs, including this mock exam from the University of Bath (Wayback Machine). One of the questions seems like it should be rather straightforward, but…
Rachel
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$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint below. Hint: There are two products on $\pi(G)$. The…
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