For questions about or involving the fundamental group.

learn more… | top users | synonyms

1
vote
1answer
39 views

Computing the fundamental group

I want to compute the fundamental group of the double Torus using the Seifert-van Kampen theorem so then I choose $U=\text{double Torus} / \{\text{point} =x_1\}$ and $V=D$ the disc. The thing is ...
0
votes
1answer
32 views

Unique homomorphisms between fundamental groups of topological spaces

Let $u:A\rightarrow B$ be a continuous map of topological spaces, $a\in A$, $b=u(a)$. How do I prove that there exists a unique group homomorphism $$u':\pi_1(A,a)\rightarrow\pi_1(B,b)$$ ...
5
votes
2answers
38 views

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology.

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology. I want to show this by showing that there does not exist any path $f$, $\forall x, y \in X$. Assume for ...
5
votes
2answers
38 views

Is the Fundamental Group of space with contractible universal cover torsion free?

Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle $K$ torsion free? We have the following presentation: $$\pi_1(K) = \langle a,b: aba = b ...
1
vote
1answer
35 views

CW structures of identification spaces, fundamental groups and universal covers

I am working on this exercise concerning the relation between some identification spaces and their CW structure and I am a bit confused. First, the question is the following: For each identification ...
2
votes
1answer
42 views

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 ...
2
votes
1answer
27 views

Infinite wedge sum of circles and a space homotopy equivalent

Given a space $G = \bigcup_{n=1}^{\infty} A_n$ where $ A_n $ is a circle $ C[ (n,0),n] \in R^{2}$ I'd like to show that its fundamental group is an infinately generated free group. So let's say $a_i $ ...
0
votes
0answers
32 views

Fundamental group and universal cover for this quotient space

For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more ...
4
votes
0answers
24 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
1
vote
0answers
17 views

Image of map of étale fundamental groups

Let $X$ be a connected (locally Noetherian) scheme and $\phi: Y \rightarrow X$ be a finite etale Galois cover of degree $d$, i.e. $Y$ is connected and there are $d$ $X$-invariant automorphisms of $Y$. ...
1
vote
1answer
27 views

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$? $\pi(X)$ denotes the fundamental group of $X$.
2
votes
0answers
59 views

Fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$

Question: Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$. $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph $T$ is the graph made of $3$ edges ...
4
votes
0answers
51 views

group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
0
votes
0answers
45 views

Computing fundamental group of the complement to three infinite straight lines, and of complement to $S^1 \cup {Z} $

Question 1: Find the Fundamental group of the complement to three infinite straight lines that have no common points in $\mathbb{R^3}$ Question 2 Compute the fundamental group of the complement of ...
0
votes
0answers
30 views

Is the complementary of a 2 codimensional set in a complex space simply connected?

It is well-known that if X is a complex manifold and $A \subset X$ an analytic set of codimension at least $2$ then $\pi_1(X)=\pi_1(X\setminus A)$. Can we generalize this equality when $X$ is a ...
0
votes
1answer
29 views

Fundamental Group Computation $\mathbb R^2 \setminus S^1$ .

I need to compute the fundamental group of $\mathbb R^2 \setminus S^1$ . First I notice this is a disconnected set, hence I will have to differentiate between two cases. If I choose the based point ...
1
vote
3answers
73 views

How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?

If A is the path connected CW complex and X is the new CW complex made by attaching a single 1-cell, is it true that the fundamental group of X is the same as that of A? I've tried to justify this ...
2
votes
0answers
24 views

Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding ...
6
votes
1answer
111 views

Is this a different proof of the fundamental group being abelian?

I have proved the fundamental group of a topological group is abelian. But I've found nowhere the similar proof as mine. Everywhere I looked up, it was done either exploiting categorical properties or ...
4
votes
0answers
29 views

How to find fundamental groups and covering spaces of $\mathbb{RP}^2\vee \mathbb{RP}^2$?

The following is an exercise I was assigned in homotopy theory. Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$. a) Find $\pi_1(X)$. b) Find the universal cover of $X$. c) Find all of its connected ...
0
votes
0answers
19 views

relation of fundamental group for different base point.

I want to know the relation between fundamental group based at $x_0$, $\pi_1(X,x_0)$ and fundamental group based at $x_1$, $\pi_1(X,x_1)$, where $X$ is a topological space. and $x_0, x_1 \in X$. What ...
4
votes
2answers
50 views

Fundamental group of Mapping Torus $M_h$: how to prove that the action is really $h_*$?

Let us work with the following setting: let $h$ be an automorphism (assume base point preserving) of a genus $g$ surface ($g>0$) to himself. $h \colon (\Sigma^g,\ast) \to (\Sigma^g,\ast)$ and ...
1
vote
1answer
19 views

How to show the constant path is the identity element in fundamental group?

Let $X$ be a topological space and $q$ is a point in $X$. Denote the fundamental group of $X$ based at $q$ by $\pi _1(X,q)$. Then how should I verify that the constant path $c_q(s)\equiv q$ is the ...
5
votes
1answer
63 views

Minimal-dimension example of (open) subset of $\mathbb{R}^n$ with trivial first cohomology but nontrivial fundamental group

As a follow-up to this question, I was wondering what dimension provides the minimal counterexample to the claims: If $U\subseteq\mathbb{R}^n$ is an open connected set with trivial $H^1(U)$, then ...
3
votes
1answer
38 views

Classification of line bundles by group homomorphisms from the fundamental group to $\mathbb{Z}_2$

Let $X$ be a path-connected manifold. Then by the classification of vector bundles, the collection of all (real) line bundles on $X$ is $$ \text{Vect}^1(X)\cong ...
3
votes
1answer
50 views

Does inclusion give an injective homeomorphism $\pi[G \cap B(a, r),a] \to \pi[G, a]$?

Let $G$ be a open, path connected subset of the plane and $a\in G$. Set $G' = G\cap B(a,r)$, where $B(a,r)$ is the open ball of center $a$ and radius $r$. Let $\gamma$ be a closed loop in $G'$ with ...
1
vote
1answer
34 views

When does triviality of $f_*$ imply null-homotopy of $f$?

I've been considering problems of the following type: Given certain topological spaces $X$ and $Y$ and a continuous function $f:X\to Y$, prove that $f$ is null-homotopic. The cases studied so ...
2
votes
1answer
49 views

Simply connected means universal-covering spaces

A covering space $p:(Y,y_0) \to (X,x_0)$ is called universal when $Y$ is simply connected (say that we restrict ourselves to path connected spaces, and locally path connected). I heard that this ...
2
votes
1answer
91 views

Past paper question without answers; could anyone provide an answer?

I apologize in advance I haven't my own approach here, it doesn't mean I haven't tried at all, I've just realized my lack of knowledge and practice in the subject. It would be great if I can get an ...
1
vote
1answer
44 views

Fundamental group - space of copies of circle $S_1$

For $n>1$ an integer, let $W_n$ be the space formed by taking $n$ copies of the circle $S_1$ and identifying all the $n$ base points to form a new base point, called $w_0$. What is $\pi_{1}$($W_n , ...
2
votes
2answers
49 views

A technical detail in the construction of a covering space: Is the projection $p$ continuous without assuming semilocal simple connectedness?

I am studying the construction of a covering space with prescribed characteristic subgroup. For simplicity, I will outline the case where the characteristic subgroup is trivial (i.e. we're dealing ...
0
votes
0answers
29 views

Fundamental group from triangulation

How do I calculate the fundamental group of surface if I have been given it in triangulated form. I have attached an example triangulation I think I may have to simplify it further to a tree and ...
4
votes
1answer
264 views

Nontrivial element in first homology of Hawaiian earring

I am stuck at the following exercise from Hatcher section 3.3: 14. Let $X$ be the shrinking wedge of circles in Example 1.25, the subspace of $\mathbb{R}^2$ consisting of the circles of radius ...
2
votes
2answers
120 views

Problem 10 of Section 1.2 from Hatcher.

Problem. Consider two arcs $\alpha$ and $\beta$ embedded in $D^2\times I$ as shown in the figure. The loop $\gamma$ is obviously nullhomotopic in $D^2\times I$, but show that there is no ...
0
votes
0answers
45 views

Using transfinite induction to compute fundamental groups

I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen. Let's ...
0
votes
1answer
25 views

Why is semi-locally simply connected defined this way?

I would like to know why we define a space $X$ to be semi-locally simply connected if $\forall p\in X \exists U\ni p: i(\pi_1(U))=0\subset \pi_1(X)$ (SLSC), where $i$ is induced by ...
1
vote
0answers
16 views

Inverse limit defining etale fundamental group.

Let $(S,s)$ be a connected scheme with geometric point $s$. In many places, I can find the etale fundamental group being defined as $$\varprojlim_{X \to S} \text{Aut}_S(X)$$ Where $X \to S$ ranges ...
0
votes
0answers
21 views

Calculating fundamental groups using retracts

I want to know if my calculations are correct. I am trying to find the fundamental group of the figure $X_n$ in $\mathbb{R}^2$ obtained by drawing $n$ lines between two points (the lines only meet at ...
0
votes
2answers
54 views

Compute explicitly a fundamental group

I want to compute the $\pi_1(X)$ where $$X=\mathbb{R}^2-(([-1,1]\times \{0\})\cup (\{0\}\times [-1,1]))$$ my only tools at the moment are the basic definitions and the fundamental group of a circle, I ...
0
votes
1answer
49 views

If an open set is evenly covered by $p$, then its open subset is also evenly covered by $p$.

If $U$ is an open set evenly covered by $p: E\to B$ and $W$ is an open set contained in $U$, then $W$ is also evenly covered by $p$. I'm trying to prove this statement, but have a difficulty. So ...
1
vote
1answer
67 views

Computation of the fundamental group of $\mathbb{C}P^n$ using induction on $n$.

Let $n\geqslant 2$, I am asked to prove the following: Proposition. $\pi_1(\mathbb{C}P^n,\cdot)$ is isomorphic to $\pi_1(\mathbb{C}P^{n-1},\cdot)$. Proof. First, let us introduce some notation: ...
1
vote
0answers
44 views

What is the fundamental group of a cone?

I am reading an article on orbifolds and it describes the cone as the quotient of unit 2-dim. disc by a finite cyclic group of rotations. But how is it's fundamental group finite cyclic?
0
votes
1answer
40 views

exercize about the foundamental group of $\mathbb{P}^n(\mathbb{R})$

Let $p$ be a point in $\mathbb{P}^n(\mathbb{R})$ and $\Sigma$ the set containing all the projective lines passing through $p$. Given $s\in \Sigma$ we can define a continous closed path (let's say ...
1
vote
0answers
30 views

On the existence of a continuous section of $\zeta\in\mathbb{S}^1\mapsto(\zeta^m,\zeta^n)\in\mathbb{S}^1\times\mathbb{S}^1$.

Let $(m,n)\in\mathbb{Z}^2$ and let define the following map: $$f:\left\{\begin{array}{ccc} \mathbb{S}^1&\rightarrow&\mathbb{S}^1\times\mathbb{S}^1\\ \zeta&\mapsto&(\zeta^m,\zeta^n) ...
1
vote
1answer
23 views

Permuted action of the ramified covering

Let $f:E\rightarrow S^2$ be a ramified covering of degree n, and let $t_1,t_2,..t_m$ be all its points of ramifications. Pick a point $t\in S^2$ distinct from all $t_i$ and connect it with the points ...
4
votes
1answer
60 views

Glueing manifolds with boundaries and Seifert-Van Kampen theorem

I've seen many times the following application of the SVK theorem: Let $M$ and $N$ two smooth $n$-manifolds ($n\ge 3$) with boundary and suppose that they have the same boundary $B$. Now, after ...
1
vote
2answers
58 views

Fundamental group of a manifold minus a submanifold

Let $X$ be a smooth $n$-manifold, with $n\ge 3$, such that $\pi_1(X)=\left<a_1,\ldots,a_m\right>$ (free group over $m$ elements) and suppose that there is an embedding: $$S^1\times ...
2
votes
1answer
37 views

Fundamental Group of Solid Octagon with Labelling Scheme

I am studying for the qualifying exams I am taking last week and am struggling with the following problem: Here's what I've tried. Say we call the left side $X$ and the right side $X'$. Pick ...
3
votes
1answer
122 views

The fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed

Determine whether the fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed is trivial, infinite cyclic, or isomorphic to the figure eight space. I found this answer: ...
-1
votes
1answer
75 views

Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...