All Questions
Tagged with free-groups fundamental-groups
27
questions
2
votes
0
answers
35
views
$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point
Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
- 430
0
votes
0
answers
44
views
Covering space of compact surface with free fundamental group
Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ?
I know that if we ...
1
vote
0
answers
66
views
How do we generate the loop $ba$ from the loops $a^2,b^2$ and $ab\ $?
In the second diagram a $2$-sheeted connected covering of the figure eight has been described. The image of the fundamental group of the covering space has the generators $a^2, b^2$ and $ab$ as ...
- 545
7
votes
2
answers
299
views
The fundamental group of closed orientable surface of genus 2 contains a free group on two generators
Let $S$ be the closed orientable surface of genus $2$. It is well known that its fundamental group is given by $$ \pi_1(S)=\langle a,b,c,d:[a,b][c,d]=1\rangle.$$
How can we show that this group has a ...
- 1,630
0
votes
1
answer
77
views
Three questions about Wikipedia's definition of Van Kampen's theorem for fundamental groups
I'm studying Algebraic Topology off of Hatcher and (unfortunately as usual) I find his definition and explanation of Van Kampen's theorem to be carelessly written and hard to follow. I happen to know ...
- 894
4
votes
0
answers
63
views
Recovering an element of a free group from its projections
Assume you have an unknown word on an alphabet with at least three letters, and you know all the words obtained by erasing each copy of some letter. Then, you can find the first letter of the original ...
- 645
1
vote
0
answers
71
views
What is the induced homomorphism of the covering map $p : Y \to X : z \mapsto z^3 - 3z$ on the fundamental groups?
Let $X = \mathbb{C} \setminus \{ \pm 2 \}$ and $Y = \mathbb{C} \setminus \{ \pm 1, \pm 2 \}$. The map
$$
p : Y \to X : z \mapsto z^3 - 3z
$$
is a 3-branched covering as given in this question of Math ...
- 11.9k
0
votes
1
answer
45
views
A question related to an induced homomorphism between two groups
Suppose $X$ is obtained by gluing two tori at a single point and let $r:\sum_2\to X$ be the retraction given by collapsing a circle around the middle of $\sum_2$ (surface of genus $2$) to a single ...
- 520
2
votes
1
answer
76
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 ($n$ ...
- 188
1
vote
1
answer
73
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)*\...
- 478
3
votes
1
answer
206
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 ...
- 186
6
votes
1
answer
252
views
In van Kampen’s theorem, what happens to the loops not in $\pi_1(U_\alpha \cap U_\beta)$?
I’ve just read the set up and the statement for van Kampen’s theorem. Here’s the version that we use in our class.
van Kampen's Theorem. Suppose we have an open cover $\left\{U_{\alpha}: \alpha \in A\...
- 2,307
1
vote
1
answer
41
views
Example of homomorphism $\underset{\alpha \in A}{\ast} \pi_1(U_\alpha) \to \pi_1(X)$, where $U_\alpha \ni x_0$ are an open cover of $(X,x_0)$.
I’m reading a passage leading up to the Van Kampen’s theorem, and I have trouble understanding the settings. Here’s the materials.
Let us consider a pointed space $\left(X, x_{0}\right)$ with an open ...
- 2,307
0
votes
0
answers
60
views
Determining the cardinality of a system of free generators for the fundamental group of $E$.
This is exercise 3 from Munkres section 84.
Let $X$ be the wedge of two circles; let $p:E\rightarrow X$ be a covering map. The fundamental group of $E$ maps isomorphically under $p_\ast$ onto a ...
user709182
2
votes
1
answer
527
views
Infinite Wedge Sum
Let $X$ be a topological space. $X$ is called Wedge of circles if $\exists \hspace{0.1cm} \left\lbrace S_{\alpha} \right\rbrace _{\alpha \in S}$ such that :
(i) $S_{\alpha} \subset X \hspace{0.2cm} \...
- 5,164