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0answers
58 views
$\pi_1$ 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 ...
2
votes
1answer
51 views
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 ...
5
votes
2answers
92 views
An intuitive idea about fundamental group of $\mathbb{RP}^2$
Someone can explain me with an example, what is the meaning why $\pi(\mathbb{RP}^2,x_0)$ is $\mathbb{Z}_2$?
consider this quotient on the disk representing the situation:
$\mathbb{RP}^2$
(sorry ...
7
votes
0answers
78 views
Finite fundamental group in the Euclidean space
Is there an example of a (path-connected) subspace of $\mathbb{R}^3$ which has a nontrivial finite fundamental group? If not, why?
4
votes
1answer
81 views
Is a subgroup of a fundamental group a fundamental group?
Let $(X,\ast)$ be a based topological space (maybe path connected or not, I don't know if this will be relevant to the solution).
Let $\pi:=\pi_1(X,\ast)$ be its fundamental group and let $H$ be any ...
2
votes
1answer
73 views
Van Kampen's Theorem with Torus and Projective Plane
I'm having some trouble finding the sets $U$ and $V$ to use for this problem.
Let $T = S^1 \times S^1$ be the torus and $P=S^2/(x \sim -x)$ the projective plane. Form the space $X$ by identifying the ...
4
votes
2answers
62 views
Conjugacy classes in the fundamental group
I want to solve the following problem (Hatcher Ch.1, problem 6):
We can regard $π_1(X,x_0)$ as the set of basepoint-preserving homotopy classes of
maps $(S_1, s_0)→(X,x_0)$. Let $[S_1,X]$ be the ...
0
votes
1answer
21 views
$Spin(N)$ $N>2$ is simple-connected
How to explain that $Spin(N)$ $N>2$ is simple-connected,
we already know fundamental group of $SO(N)$ $(N>2 )$is $Z/2$,and $Spin(N)$ is $SO(N)$
nontrivial double covering.
2
votes
0answers
33 views
Number of sheets of Covering space of $\mathbb{T}^2$ after transformation by $SL(2,\mathbb{Z})$
Suppose you have a subgroup of $H\subset\pi_1(\mathbb{T}^2)=\mathbb{Z}\times\mathbb{Z}$, $H=\text{span}\langle u,v\rangle$. If you have an element $G\in SL(2,\mathbb{Z})$, do the covers of ...
1
vote
1answer
43 views
Induced homomorphisms in a universal covering map
I don't understand what happens in the induced homomorphism of a universal covering map.
A covering map $p:\tilde X \to X$ is universal if $\tilde X$ is simply connected, so the fundamental group ...
0
votes
1answer
28 views
Question about the fundamental group of simplicial complex.
Suppose we have a simplicial complex G that is finite connected.
(1)The fundamental group of G is finite;
(2)The universal cover of G is compact.
Question:
Can (1) implies (2)?
Thanks.
2
votes
1answer
39 views
Universal cover modulo the monodromy action
Let $\tilde{X} \to X$ be the universal cover of a connected, locally path-connected and semi-locally simply connected topological space $X$. Is it always true that the orbit space
$$ \tilde{X} \;/\; ...
2
votes
0answers
36 views
How do we check if a covering of an orbifold is a manifold?
Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
3
votes
1answer
48 views
Fundamentalgroup of $\mathbb{R}P^2$
i have to compute the fundamentalgroup of the projective plane $\Bbb{R}P^2$ with Van-Kampen Theorem. Therefore i use the fundamentalpolygon given in Projective Plane.
I get for the presentation of the ...
3
votes
0answers
71 views
Classification of fundamental groups of non-orientable surfaces
I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$.
I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
0
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1answer
47 views
why product of two path homotopy class is not always defined?
This is a question regarding lack of formation of Fundamental Group, could any one give an easy example why the product of two path homotopy class is not defined in a space $X$?
Thank you.
1
vote
1answer
81 views
homotopies of paths in the cylinder
Well I tried to imagine all this paths in my mind ( actually I used a glass and wool) to define informally the homotopy between these four paths, I think that they are homotopic, but how can I prove ...
12
votes
2answers
200 views
Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?
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 ...
3
votes
0answers
38 views
isomorphisms of $\pi_{1}(T^2, x_{0})$ with itself.
I1m studying fundamental group and its relation with covering maps, I was thinking about an exercise:
every isomorphism of $\pi_{1}(T^2, x_{0})$ with itself is induced by a homomorphism $f:T^2 ...
1
vote
2answers
42 views
a proof concerning fundamental group and lifting of paths
Well, I put the theorem $54.3$ only to show that $\phi$ is well defined.
I'm not sure why $\phi([f])= e_1 $ only by definition. Because to check that, I have first to lift $f$ and then compute $
...
0
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1answer
39 views
Two functions from $B^{2}$ to $S^{2}$ with this conditions are equal or antipodal at one point
I have been trying to solve this exercise from the book Fundamental Group and Covering Spaces written by Elon Lages Lima, chapter 3. It says that given $f,g : B^{2} \to S^{2}$
continuous such that ...
2
votes
1answer
80 views
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 ...
2
votes
1answer
39 views
What do we know about smooth families over the open unit disc
Let $B=B(0,1)$ be the open unit disc in $\mathbf C$. Let $X\to B$ be a smooth projective morphism of complex manifolds with $X$ connected.
What can we say about $X$?
For instance, if $X\to B$ is of ...
3
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0answers
72 views
The fundamental group of the union of three convex open subsets of $ \mathbb{R}^n$.
I have to prove that the fundamental group of the union of three open convex subsets of $\mathbb{R}^n$ is trivial or $\mathbb{Z}$. I can show that it has only one generator, but I can't prove that if ...
2
votes
0answers
54 views
The fundamental group of an open set in $\mathbb{R} ^n$ does not have nilpotent elements.
I am studying a little of basic algebraic topology and I thought that this statement could be true. If you have an open connected set $U \subset \mathbb{R}^m$ and a loop $\gamma$ that is not ...
0
votes
1answer
56 views
Abelianization of the $\mathbb R \mathbb P^2$#$\mathbb R \mathbb P^2$
I know that the fundamental group of $\mathbb R \mathbb P^2$#$\mathbb R \mathbb P^2$ is $<a,b|a^2b^2>$.
When we abelianize this we have this presentation: $Ab(\pi_1(\mathbb R \mathbb ...
9
votes
1answer
262 views
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.
...
10
votes
2answers
127 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?
0
votes
1answer
101 views
Fundamental group of this space
Based on this question: What is the homology groups of the torus with a sphere inside?
I'm trying to find the fundamental group of this space using the Seifert–van Kampen theorem. If $U$ is the torus ...
1
vote
1answer
144 views
Prove that the spaces have the same homotopy type
This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima.
"Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...
3
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0answers
61 views
A proof of simply connectedness of a symplectic quotient
Let $\rho$ be a unitary representation of a torus $G$ on $\mathbb{C}^n$. The action of $\rho$ is Hamiltonian with a moment map $\mu:\mathbb{C}^n \to \mathfrak{g}^*$. Here $\mathfrak{g}^*$ is the dual ...
2
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1answer
112 views
Easy papers on fundamental groups (for beginners)
I'd like to read some papers concerning fundamental groups, for example, papers written to explain some basic facts about homotopy explicitly for undergraduate students.
All the papers I have ...
0
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1answer
130 views
Problems to understand Van Kampen theorem
I have problems to understand the Seifert-Van Kampen theorem when $U, V$ and $U\cap V$ aren't simply connected. I'm going to give an example:
Let's find the fundamental group of the double torus $X$ ...
5
votes
1answer
196 views
How can I prove $\pi_1(M)=\pi_1(M-\{q\})$?
If $M$ is a connected topological manifold with dimension $n\ge 3$, let $q\in M$. How can I show that $\pi_1(M)=\pi_1(M-\{q\})$?
As Neal said, since M is locally connected, M is path connected, then ...
0
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2answers
197 views
Dunce hat is simply connected
I'm trying to prove that the dunce cap is simply connected via Seifert- Van Kampen Theorem. I choose to be my open sets $U$ and $V$ the open disk and the punctured surface below, then $U\cap V$ is the ...
1
vote
1answer
52 views
What kind of information is given by the n-th homotopy group for n>=2?
How i see this the homotopy group of order 1 is giving information about the "holes" in a topological space. In that way what kind of information is giving the homotopy group of order 2 or n in ...
0
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0answers
79 views
If we know the fundamental group of $X$, we know the fundamental group of $X/\sim$
I'm trying to find the fundamental group of the cone, and this question appear on my mind:
Since we can defined the cone as $CX=X\times I/I\times \{1\}$, what we can say about $CX$? If we know the ...
0
votes
3answers
84 views
Fundamental group of $\mathbb R^2$ without a closed semi-line
Intuitively the fundamental group of the $\mathbb R^2$ without a closed semi line is the trivial group, but I don't know how to prove it.
Any ideas? thanks
1
vote
1answer
110 views
The fundamental group of the $ \mathbb R^3$ without a axis
I'm trying to find the Fundamental group of the $\mathbb R^3$ without the axis $z$. Intuitively, It's easy to realized that this fundamental group is the same of the circle, i.e., $\mathbb Z$. I don't ...
1
vote
2answers
83 views
$h_*$ is the trivial homomorphism, the $h$ is homotopic to a point
Let $h:S^1\to X$ a continuous map. If $h_*:\pi_1(S^1)\to \pi_1(X)$ is the trivial induced homomorphism, then $h$ is homotopic to a point.
I'm starting to study fundamental groups and induced maps by ...
5
votes
4answers
305 views
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 ...
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0answers
88 views
How do I prove this function is not continuous?
Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$.
The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
5
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0answers
73 views
Normal subgroups of the fundamental group of a non-orientable surface.
Let $N^2_g$ be a non-orientable closed genus $g\geq 2$ surface. Is there a way to explicitly list the normal subgroups of $\pi_1(N^2_g)$ in terms of generators and relations? I am interested in ...
0
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1answer
149 views
Calculating the Fundamental group of $\Bbb R P^2$
The fundamental group of $\Bbb R P^2$ is $\Bbb Z \times \Bbb Z$.
I cannot understand why though, since $\Bbb RP^2$ is a disc with a Mobius strip and the disc is contractible so wouldn't it have ...
3
votes
2answers
358 views
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?
0
votes
1answer
120 views
Covering space and Fundamental group
Let $p:E\to X$ be a covering space and $\pi_1(E)$ be a fundamental group of $E$.
Can you give me a recept for calculating a fundamental group $\pi_1(X)$ (may be for some special cases)?
Thanks a lot!
...
3
votes
1answer
69 views
etale fundamental group of a product
Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism
$\pi_1(X \times ...
