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Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

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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 ...
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2answers
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Fundamental group of a torus with points removed

Question 5.33 from "Topology and its Applications" by Baesner is to compute the fundamental group of the torus ($T^2$) with $n$ points removed. I can "see" in my mind that if we remove one point we ...
15
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1answer
960 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$? [duplicate]

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
10
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1answer
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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 ...
22
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4answers
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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 ...
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2answers
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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....
36
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3answers
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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 ...
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1answer
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Fundamental groups of Grassmann and Stiefel manifolds

Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?
12
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2answers
532 views

Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
3
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2answers
851 views

intuition on the fundamental group of $S^1$

I am familiar with the proof that the fundamental group of the unit circle $S^1$ is $\mathbb Z$, yet I couldn't develop intuition for why it is true. For example, why would I fail if I try to find ...
8
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1answer
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Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
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1answer
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Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
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2answers
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Fundamental group of $\mathbb{R}^3$ minus trefoil knot

Let $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van ...
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0answers
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Subset of $\mathbb{R}^3$ with an element of finite order in its fundamental group

Is there a subset of $\mathbb{R}^3$ with an element of finite order (not the identity!) in its fundamental group? I think the real projective plane is such a subset as its fundamental group is ...
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7answers
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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 ...
18
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4answers
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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 ...
8
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1answer
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The fundamental group of a topological group is abelian [duplicate]

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I think,...
14
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2answers
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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 ...
15
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1answer
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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 $a$...
13
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3answers
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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?...
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Is the fundamental group of a compact manifold finitely presented?

Let $X$ be a connected compact smooth manifold. If $X$ is boundaryless, we can choose a Riemannian metric for $X$ so that $\pi_1(X)$ acts geometrically (ie. properly, cocompactly, isometries) on the ...
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2answers
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$\pi_1$ and $H_1$ of Symmetric Product of surfaces

Let $X=Sym^d(\Sigma_g)$ be the d-fold symmetric product of a genus-g surface, $d\ge 2$. Is there / what is a (quick simple) way to see that $\pi_1(X)$ is abelian? The link in the comments (Ozsvath-...
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1answer
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Fundamental Group is free on infinite generators.

This is question 16 of section 1.2 in Hatcher's Algebraic Topology. I have to show that the fundamental group of the space $X$ is free on an infinite number of generators. So here is my approach. ...
7
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1answer
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The fundamental group of $U(n)/O(n)$

Since $O(n)$ is a subgroup of $U(n)$, we can consider the quotient space $L_n=U(n)/O(n)$. The quotient space is homotopic to the ($n$-th) Lagrangian Grassmannian, and it is known that its fundamental ...
6
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1answer
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Fundamental group of a CW complex only depends on its $2$-skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1(X)$ only depends on the $2$-skeleton of $X$ for any CW complex $X$. I am very ...
3
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1answer
841 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface (...
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2answers
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Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
3
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1answer
871 views

If $Y$ is path-connected, then there is only one homotopy class of maps $[0,1] \to Y$

I have this exercise: If Y is path-connected, show that there is only one homotopy-class of continuous functions from $[0,1]$ to Y. My attempt: What I need to show is that if I have two ...
5
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1answer
176 views

A topological space $X$ who can be $drawn$, whose fundamental group is $S_3$

There exists an example of a topological space $X$ who can be $drawn$, whose fundamental group is $S_3$? I know that every group is the fundamental group of a topological space, but I need a concrete ...
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2answers
258 views

Extending to a disc means fundamental group is trivial

Working on this problem. If any $f:S^1\rightarrow X$ extends to a $F:D^2\rightarrow X$, then $\pi_1(X, x_0)$ is trivial. We can turn any loop $f^\prime:I \rightarrow X$ in $\pi_1(X, x_0)$ into ...
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1answer
389 views

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ [duplicate]

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ where $(f\otimes g)(s) = f(s) \cdot g(s)$ where $\cdot$ is the group operation on the topological group $G. $ This is a question from ...
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1answer
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Does there exist a CW complex with prescribed fundamental group and trivial higher homology?

In this question, the OP asks whether it is possible to find a CW complex with prescribed homology groups and fundamental group. In my (partial) answer, I point out that by taking the wedge sum of ...
5
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1answer
388 views

Why does Van Kampen Theorem fail for the Hawaiian earring space?

The Hawaiian earring space has notoriously complicated fundamental group, and is essentially not as simple as the wedge sum of countably many circles whose fundamental group is straightforwardly given ...
2
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1answer
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Associativity of concatenation of closed curves from $I$ to some topological spaces $X$

I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
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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. ...
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2answers
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Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?

I am curious if there is a decent "bare hands" proof that the fundamental group of $S^1$ is $\mathbb Z$ that does not invoke covering space theory. One must show two claims. First, that $f(t)=e^{2\pi ...
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3answers
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Fundamental group of projective plane is $C_{2}$???

I just recently know that there are topology with finite nontrivial fundamental group (homotopy curve). I just can't wrap my mind around it at all. If you have a curve, and somehow cannot shrunk it ...
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2answers
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Prove every map from the projective plane to the circle is nullhomotopic

Prove that every continuous map $f:P^2\to S^1$, where $P^2$ is the projective plane, is nullhomotopic. I think I need to use the fact that $\pi_1(P^2) = \mathbb{Z}/2\mathbb{Z}$ and covering space ...
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1answer
67 views

Fundamental group of $Spin^c(V)$

In picture below, Spin(V) is Spin group. In the last paragraph , it wants to show $\pi_1(Spin^c(V))=\pi_1(S^1)$. But I almost don't know it. First, where is the $S^1$ in ? For 'a homotopically ...
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3answers
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Homotopy equivalent spaces have homotopy equivalent universal covers

A problem in section 1.3 of Hatcher's Algebraic Topology is Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. ...
6
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1answer
459 views

Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over $\...
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1answer
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Need help understanding statement of Van Kampen's Theorem and using it to compute the fundamental group of Projective Plane

I'm using the statement from Hatcher. I really don't understand the statement of the theorem, let alone the proof, and I especially don't understand what the normal subgroup $N$ generated by $i_{\...
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1answer
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If 2 spaces are homotopy equivalent, then their fundamental group is the same

If 2 spaces are homotopy equivalent, then their fundamental group is the same Is this true ? Let $f:X\rightarrow Y$$\quad$$g:Y\rightarrow X$ s.t. $f\circ g\simeq id_Y$$\quad$$g\circ f\simeq ...
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2answers
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fundamental group of $U(n)$

Is my logic correct? $f:U(n)\rightarrow U(1)$ defined by $f(A)=\det A$ is a group homomorphism so that the induced homomorphism $f^{*}: \pi_1(U(n))\rightarrow \pi_1(U(1))$ will be an isomorphism, ...
3
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1answer
470 views

Base Point Homotopy vs Free Homotopy Example

Does there exists a space X, along with two loops $f,g$ based at $x_0 \in X$ such that $f$ and $g$ are freely homotopic and not "based point" homotopic? Two loops are freely homotopic if there ...
7
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3answers
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Fundamental group of Hawaiian earring

I am trying to understand how the fundamental group of the infinite shrinking wedge of circles is $G=\prod_{i=1}^\infty\mathbb{Z}$. I understand that it is something more than $H=\bigoplus_{i=1}^\...
6
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2answers
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Why is the fundamental group a sheaf in the etale topology?

In this paper by Minhyong Kim on p5, there is a variety $X$ defined over $\mathbb{Q}$, $G = \pi_1(X(\mathbb{C}),b)$ the topological fundamental group of the associated complex algebraic variety, and $...
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3answers
810 views

Fundamental group of complement of circle with a line passing through it.

Let $X= S^1 \cup y$-axis in the $xy$-plane. Compute the fundamental group of $\mathbb{R}^3 - X$ Edit: To clarify, $S^1 =\{(x,y)|x^2 + y^2 = 1\}$ is the unit circle in the $xy$-plane, so that the $y$...
8
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3answers
763 views

Finding a space with given homology groups and fundamental group

I started thinking a couple days ago about the example below, and it led me to ask the following question: How (or when?) can we build a space (let's say a CW complex) with given homology groups and ...
8
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2answers
546 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...