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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
43 views

Degree of a map $f:S^1 \to GL(N,\mathbb{C})$

This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $...
TRay's user avatar
  • 1
0 votes
1 answer
46 views

Prove that a square without one inner point is not simply connected

I understand that this is a consequence of Brouwer's theorem. For a circle without an inner point, we have that the circle's mapping to itself does not contract, i.e. if $f = \mathrm{id}\colon S^1 \to ...
jhonnnnnny's user avatar
0 votes
1 answer
54 views

How will the map be described?

Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\...
Hope's user avatar
  • 85
2 votes
3 answers
65 views

Fundamental group of simplicial space

This question asks the same as mine, but it was unsuccessful in getting an answer, so I try again. For context I am reading Weibel's K-book and am struggling with proposition 8.4 which computes $K_0$...
DevVorb's user avatar
  • 1,255
-1 votes
1 answer
68 views

Fundamental group of cyclic group $Z_n$ [closed]

How to calculate the fundamental group of discrete group like the cyclic group $Z_n$? Physically, it seems that $\pi_1(Z_2)$ will be trivial, as there can be no vortex structure for $Z_2$ symmetry. ...
ZJX's user avatar
  • 261
3 votes
0 answers
31 views

Existence of special transversal on foliation

This is a somewhat technical question about a line in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Program in the proof of the structure theorem, Theorem 8.3. We ...
subrosar's user avatar
  • 4,634
2 votes
0 answers
67 views

Question regarding the fundamental group of $SU(n)$

I want to prove that $\pi_1 (GL(n,\mathbb{C})) \simeq \mathbb{Z}$. To do so, i alredy proved that $GL(n,\mathbb{C})$ and $U(n)$ have same homotopy type and that there is and homeomorphism between $U(n)...
Ggstal's user avatar
  • 21
0 votes
0 answers
46 views

some detail in the proof of Invariance of domain theorem in Munkres

the above is from Munkres Topology according to it $ f : U\to S^2$ is continuous and injective and $B$ any closed ball contained in $U$ and $a , b$ two points of $S^2-f(B)$.it says because the ...
Davood Karimi's user avatar
0 votes
0 answers
35 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 ...
Serge the Toaster's user avatar
2 votes
1 answer
41 views

Hatcher Theorem 1B.8 / Proposition 1B.9 concluding that a map inducing the identity is homotopic to the identity?

In introducing the concept of $K(G, 1)$, Hatcher's Algebraic Topology proves the theorem that the homotopy type of a CW complex $K(G, 1)$ is uniquely determined by $G$, by citing a proposition and ...
I Eat Groups's user avatar
0 votes
1 answer
88 views

Homotopy classes of paths on $\mathbb{R}^2 \setminus\{0\}$

Consider the plane with a point removed $M=\mathbb{R}^2\setminus\{0\}$. Let $p_1,p_2\in M$ be two points. I want to understand how to classify the homotopy classes of paths joining $p_1$ and $p_2$. ...
Gold's user avatar
  • 26.4k
0 votes
1 answer
91 views

fundamental group of a sphere with 2 handles

Prove that the fundamental group of a sphere with 2 handles contains a free group of 2 generators. Intuitively, it is clear that we can cut a figure homotopically equivalent to a bouquet of two ...
yehehhd's user avatar
  • 109
0 votes
1 answer
64 views

What exactly is the proof of onto here?

I have been reading the solution of the following question: We can regard $\pi_1(X, x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S^1,s_0) \to (X, x_0).$ Let $[S^1, X]$ be the ...
Brain's user avatar
  • 989
1 vote
1 answer
81 views

Proving the fundamental group of circle is infinite.

I am trying the prove that fundamental group of circle is infinite by using the concept of degree of loops. Here I am giving the details of what definitions I am using here. $$S^{1}:=\{ z \in \mathbb{...
Arshdeep Sandhu's user avatar
1 vote
0 answers
77 views

the fundamental group of double torus( $T\#T$) is not abelian

I find this proof from Munkres Topology lacking rigour and a lot of precision. can someone write down the equations that have been mentioned in the proof and probably provide some feeling or intuition ...
Davood Karimi's user avatar
0 votes
1 answer
57 views

Prove that $A$ is a deformation retract of $X$

Let $X$ be the comb space given by $$ \begin{align} X ={} &\bigl\{ (x,y) \in \mathbb{R}^2 \mid 0 \leq y \leq 1 \text{ and } (x=0 \text{ or } x=\tfrac{1}{n} \text{ for some} \in \mathbb{N} \bigr\...
jasmine's user avatar
  • 14.4k
1 vote
0 answers
60 views

Fundamental group of $S^2\times S^2$ under a quotient

Let $$S^2 = \{(x,y,z)\in\mathbb{R}^3 : x^2 + y^2 + z^2 = 2\}$$ and $\tilde{X}=S^2\times S^2\subset\mathbb{R}^6$. Define an equivalent relation $\sim$ on $S^2\times S^2$ setting $$(x_1,y_1,z_1,x_2,y_2,...
Eric Vaz's user avatar
  • 397
0 votes
0 answers
27 views

Fundamental group of the complement of a quadric cone with two tangent planes

Let $X=\mathbb{C}^3$, $Q=V(x^2-yz)$, $H_1=V(y)$, $H_2=V(z)$. The hyperplanes $H_i$ are tangent to $Q$ along a line. Let $U$ be the complement of $Q\cup H_1\cup H_2$. Is the fundamental group $\pi_1(U,\...
Sergey Guminov's user avatar
0 votes
2 answers
42 views

Homotopy between the trivial element of $\pi_1(S_1)$ and a single loop around the circle

I am new to this topic of homotopy and the fundamental group. I have just read the proof that the fundamental group of $S_1$ is isomorphic to $\mathbb{Z}$. There's something that has been bugging ...
Mitchell Butovsky's user avatar
0 votes
1 answer
85 views

Fundamental group of $\mathbb{R}^{n}$ minus a circle [closed]

I know that $\mathbb R^3-\text{circle}$ has a fundamental group equal to $\mathbb Z$ and I was wondering if some things could be said about $\mathbb R^n-\text{circle}$ for $n>3$. In particular, I'd ...
Maxime CAILLEUX's user avatar
2 votes
0 answers
41 views

Characterization of the center of the fundamental group

As shown in this post: Center of fundamental group We know that for any homotopy from the Identity to itself $F: X\times I \rightarrow X$ For any $x_0 \in X$, the class of the loop $F(x_0, \cdot ):I\...
GC.'s user avatar
  • 21
1 vote
1 answer
107 views

Is $f$ homotopic to $g?$ Yes/ No

let $S^1$ be the unit circle of complex plane and let $f, g:S^1 \to S^1$ the map $f(z)=z$ and $g(z)=z^2$ Is $f$ homotopic to $g?$ My attempt : Yes, I think Take the map $F : S^1 \times [0,1] ...
jasmine's user avatar
  • 14.4k
1 vote
0 answers
48 views

Construction of Covering space by Galois correspondence

We know, If $X$ is path connected,locally pathconnected,semi locally simply connected then $X$ has a universal cover $\overline X$.Now, By Galois Correspondence for any subgroup $H$ of $π_{1}(X)$ we ...
Nope's user avatar
  • 1,222
4 votes
0 answers
62 views

Constructing a covering space of the wedge of two circles correponding to a subgroup of the fundamental group

I am learning covering spaces for the first time and came across the following problem: Describe the covering space of the wedge of two circles that corresponds to the subgroup of $\pi_1(S^1\vee S^1) ...
Eric Vaz's user avatar
  • 397
0 votes
1 answer
166 views

Generalized wedge sums of multiple $S^n$

Fix $n, m\in\mathbb{N}$. We use $D_n$ to denote the disjoint union of two $(n-1)$-dimensional closed disks. For each $1\leq i \leq m$, we use $S(n, i)$ to denote $S^n$. Then clearly $D_n$ can be ...
Sanae's user avatar
  • 321
0 votes
0 answers
54 views

Give 6 topological spaces satisfying this

As an excercise, I have to give 6 compact, Hausdorff and path connected topological spaces ${Z_k}_{k=1}^6$ such that pairwise are not homotopic and satisfying that $|\pi_1 (Z_k) | \leq 5$. My question ...
Daniel García's user avatar
3 votes
1 answer
178 views

A complete exercise of Algebraic topology [closed]

My teacher of Algebraic topology has shared exams from past years and this exercise appears on them: Consider the following sets on the Euclidean plane $\Bbb{R}^2$: $$X_1 = \{(x,y) \in \Bbb{R}^2 : x^...
Superdivinidad's user avatar
1 vote
0 answers
84 views

Computing fundamental groups of some figures

As an excercise of my Algebraic topology lessons, I have to compute the fundamental group of the following figures: There is no any information, only the drawing, so I am a little bit lost on how to ...
Daniel García's user avatar
1 vote
2 answers
139 views

Computing the fundamental group of a quotient space

I have been reading some notes of former students of my Algebraic topology lessons and I have seen this excercise: Let denote $D^2:= \{(x,y) \in \Bbb{R}^2 : x^2 + y^2 \leq 1 \}$ and $S^1=\partial B^2$...
Superdivinidad's user avatar
0 votes
1 answer
44 views

Prove that there is no function $f: B[0,1] \to \mathbb{S}^1$ such that $\left.f\right |_{\partial B[0, 1]} = id$

I have to prove that there is no function $$f: B[0,1] \to \mathbb{S}^1 $$ such that $$\left.f\right |_{\partial B[0, 1]} = id \tag{$\star$}$$ I want to use the concept of fundamental groups (however ...
Grant's user avatar
  • 61
3 votes
1 answer
47 views

Computing the fundamental group of a union of subspaces

If $x, y \in \Bbb{R}^n$, we denote by $[x,y] := \{ (1-t) \, x + t \, y : t \in [0,1] \subset \Bbb{R} \}$ Let denote the following subspaces of $\Bbb{R}^2$: $$ X_1 := \{ x \in \Bbb{R}^2 : ||x||=1 \} \\ ...
Superdivinidad's user avatar
0 votes
2 answers
58 views

Fundamental group of logarithmic surface

I am trying to determine the fundamental group of the following parametrized surface: $$ X(r,\theta) = (r\cos\theta,r\sin\theta, ln(r^2)),$$ where $r\in(0,+\infty)$ and $\theta \in [0,2\pi)$. It can ...
Rainbow's user avatar
  • 151
1 vote
0 answers
181 views

Non-normal coating of the Klein bottle

(a) Provide an example of a non-normal connected covering map $p:(\tilde X, \tilde x_0) \to (K, x_0)$ where $K$ is the Klein bottle. (b) Choose $x_0 \in K$ and $\tilde x_0 \in p^{-1}(x_0)$. State what ...
Andreadel1988's user avatar
6 votes
3 answers
199 views

If $\pi_1(X)\simeq G\times H$ then $X\cong Y\times Z$ such that $\pi_1(Y)\simeq G$ and $\pi_1(Z)\simeq H$

Suppose $X$ is a topological manifold such that $\pi_1(X)\simeq G\times H$ for some nontrivial groups $G$ and $H$. Then can I always find topological manifolds $Y$ and $Z$ such that $X\cong Y\times Z$ ...
one potato two potato's user avatar
3 votes
2 answers
185 views

Recovering a topological space from its fundamental groupoid [closed]

Given only the fundamental groupoid of a topological space X, we can recover the underlying set of X since objects of the groupoid (as a category) are precisely the elements of X. I would like to know ...
toby flenderson's user avatar
0 votes
0 answers
62 views

Verification of elementary proof of $\pi(S^1)$ being isomorphic to $\mathbb{Z}$

I have just begun learning algebraic topology, and was trying to find an elementary proof of the fact that the fundamental group of $S^1$ is isomorphic to $\mathbb{Z}$. I think I now have an answer, ...
QED's user avatar
  • 867
2 votes
2 answers
86 views

$S^1$ has no deformation retracts other than itself

I am trying to prove that if $A \subset S^1$ is a deformation retract of $S^1$, then $A=S^1$. $A$ being a deformation retract of $S^1$ means that there exists a continuous map $r \colon S^1 \to A$ ...
David's user avatar
  • 95
0 votes
1 answer
36 views

A specific cell complex

We construct a cell complex by attaching the boundary of a two dimensional disk $D$ to $S^1$ by $z\to z^n$ ($n>2$). This cell complex seems to be closed, compact and connected. But it's ...
max101's user avatar
  • 1
3 votes
1 answer
117 views

What is the value of homotopy group $\pi_1(SU(2)\times U(1))$

Question background: In particle physics, the weak interaction symmetry group is described by $G=SU(2)\times U(1)$, which is spontaneously broken into $H=U(1)_{em}$. ($U(1)_{em}$ is the ...
Daren's user avatar
  • 215
0 votes
1 answer
104 views

Can two paths with different end points be homotopic to each other?

I'm watching this video about fundamental groups. Around minute 3:20 if I have understood it well, the lecturer says that two paths with different end points can't be homotopic, right? If so, why is ...
user32415's user avatar
0 votes
1 answer
58 views

Map Without Lift

I want to show if $M$ is compact orientable surface of genus $2$, then there exists continuous map $f : M \rightarrow S^1$ with no lift to continuous map $\overline{f} : M\rightarrow \mathbb{R}$. I ...
Laurence PW's user avatar
1 vote
0 answers
43 views

simply connected Lie groups are isomorphic iff locally isomorphic

I am reading a lecture note on Lie groups, and in the note it states that upon "standard topological considerations", two simply connected Lie groups $G,H$ are locally isomorphic if and only ...
Kai's user avatar
  • 11
1 vote
1 answer
105 views

What if $X$ is not path connected?

I know that the following statement is true: $H^1(X, \mathbb Z)$ is isomorphic to $\operatorname{Hom}(\pi_1(X, x_0), \mathbb Z),$ for any path-connected pointed space $(X, x_0).$ My question is what ...
Brain's user avatar
  • 989
2 votes
1 answer
103 views

$\Sigma \mathbb{R}P^n$ and $S^2 \vee S^3 \vee ... \vee S^{n+1}$ are not homotopy equivalent

I want to use all the material for my algebraic topology class since I'm quite lost in it... For that, I want to show that $\Sigma \mathbb{R}P^n$ and $S^2 \vee S^3 \vee ... \vee S^{n+1}$ are not ...
marilou64's user avatar
  • 464
2 votes
0 answers
83 views

Fundamental group and base points.

Assume that $ N $ is a path-connected smooth manifold. $ \alpha,\beta:\mathbb{S}^1\to N $ are two loops. $ x_1\in\alpha $ and $ x_2\in\beta $ are two points on the loops $ \alpha $ and $ \beta $. ...
Luis Yanka Annalisc's user avatar
0 votes
0 answers
111 views

Calculating fundamental group of Klein bottle

To calculate the fundamental group of the Klein bottle, $X$, we can find path connected open sets, $U$, $V$,and $U \cap V$, satisfying $U\cup V= X$. I know we can choose $U$ and $V$ as follows:[![$U$ ...
Gunt Ryumet's user avatar
1 vote
2 answers
78 views

How to judge whether a loop in $ SO(3) $ is trivial or not.

Consider $ SO(3) $, I have a loop defined as follows $$ f(\theta)=(n(\theta),m(\theta),n(\theta)\times m(\theta)), $$ where for $ \theta\in[0,2\pi] $, \begin{align} n(\theta)&=(\cos\phi\cos\theta,\...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
96 views

$\pi_1(F_4)$ contains $\pi_1(F_{10})$ as a subgroup of index $3$

Let $n$ be a natural number. We denote by $F_n$ an orientable, compact & connected surface of genus $n$. I now have to show that $\pi_1(F_4)$ contains $\pi_1(F_{10})$ as a subgroup of index $3$. ...
Minerva's user avatar
  • 153
2 votes
2 answers
95 views

Is there a space with fundamental group isomorphic to $\mathbb{R}$? [duplicate]

I am studying basic algebraic topology and all examples of fundamental groups $\mathbb{Z}, \mathbb{Z^n \times \mathbb{Z}} / n \mathbb{Z} $, $\mathbb{Z}*\mathbb{Z}$ etc are somewhat discrete. I don't ...
Mahammad Yusifov's user avatar
0 votes
1 answer
71 views

Fundamental group of this subset of $\mathbb{C}^2$

How to compute the fundamental group of the following topological space? $$\{(z_1,z_2)\in\mathbb{C}^2\ |\ z_1\neq0,\ z_2\neq0,\ z_1\neq z_2\}$$ I am having no idea..
likeeatingoctopus's user avatar

1
2 3 4 5
35