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.

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0answers
23 views

Bijection Between Maps of Covering Spaces and Maps of Fundamental Groupoid Covers

I'm reading May's Concise Course in Algebraic Topology. He states: My question is about this last corollary. How does the bijection $\text{Cov}(E, E') \to \text{Cov}(\Pi(E), \Pi(E'))$ immediately ...
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1answer
14 views

Question on induced homomorphism involving translation maps

I'm working with Viro's textbook on topology, and I stuck on this exercise on induced homomorphisms: Where $ T_s : \pi_1(X,x_0) \to \pi_1(X,x_1) : [\alpha] \mapsto [s^{-1}\alpha s] $ -- a translation ...
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4answers
141 views

Fundamental group of the total space of an oriented $S^1$ fiber bundle over $T^2$

Let $M$ be the total space of an oriented $S^1$ fiber bundle over $T^2$. Can we show the fundamental group of $M$ is nilpotent? More generally, how can we calculate the fundamental group of $M$ ...
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Intuition behind coverings of $S^1 \vee S^1 $

I was recently studying Algebraic Topology reading Hatcher, and came across a table of diagram that talks about covering spaces of $S^1 \vee S^1$ on page 58. I don't really understand how these are ...
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1answer
40 views

Cell structure of a torus with an open disk removed

I'm reviewing Algebraic Topology and this is an old homework "Viewing the torus T as usual as the square $[-1,1]^2$ with opposite sides identified, let $X$ be obtained from T by removing the open ...
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1answer
20 views

Showing a inequality in Poincaré-Bohl (Elon Lages Lima)

In the book "Fundamentals groups and covering spaces" by Elon Lages Lima: I can't prove equality. I only get the following (and similar results) $|f(x)-g(x)|=|f(x)-tf(x)+tf(x)-g(x)+tg(x)-tg(...
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1answer
54 views

Homotopy between paths

Consider $f,\overline{f}:\left(S^1, s_0\right) \to (X,x_0)$ two paths in $X$ such that $\overline{f}=f \circ r$, where $r: S^1 \to S^1$ is the reflection around the $1$-space with $x_2 = 0$. I would ...
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0answers
34 views

Surface identification and fundamental group.

I am trying to identify and find the fundamental group of the surface $M,$ which is constructed by removing the shaded middle section from the surface below and replacing it with a four-holed Klein ...
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1answer
58 views

A lift of a map from the projective plane to itself

Let g: $RP^2 \mapsto RP^2$ be a map such that $g_*$ the induced map on the fundamental group is not the zero map. We need to prove that g can be lifted to a map $T:S^2 \mapsto S^2$ such that $T(-x)= -...
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1answer
53 views

determining the quotient group in Mayer-Vietoris sequence

I am having trouble to determine the quotient group in the following Mayer-Vietoris sequence. I know this problem in Hatcher exists here but my question is not to have a solution (because I do have ...
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0answers
55 views

Action of the Fundamental Group of Klein Bottle over $\mathbb{R}^{2}$

It is well know that the Fundamental Group of the Klein Bottle is defined by $$G=BS(1,-1)=\langle a,b:bab^{-1}=a^{-1}\rangle$$ An explicit description can be obtaned by define $G$ as the group of ...
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2answers
293 views

Fundamental group of Klein Bottle

It is well know that the fundamental group of the Klein Bottle $G$ is defined by $$G=BS(1,-1)=\langle a,b: bab^{-1}=a^{-1}\rangle.$$ I know, for example that $BS(1,2)$ can be defined as the group $$...
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1answer
28 views

Prove that a fundamental group is equal to a semidirect product. [duplicate]

Let $X$ be a connected space that admits universal covering $E$ and $f:X \to X $ an homeomorphism . Now let’s call $Y=(X\times [0,1])/\sim$ where $\sim$ is the relation generated by $(0,x)\sim(1,f(x))$...
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1answer
56 views

What am I misunderstanding about this isomorphism?

From Rotman's Algebraic Topology: If $K$ is a connected simplicial complex with basepoin $p$, then $\pi(K,p) \simeq G_{K,T}$, where $T$ is a maximal tree in $K$ and $G_{K,T}$ represents the group ...
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43 views

Can we find an embedding from $T^n$ to $(\mathbb C^*)^n$ which is not isotopy/homotopy to the standard one?

Suppose there are two topological embedding $f^k: T^n=(S^1)^n\to (\mathbb C^*)^n$ ($k=0,1$). They respectively induce homomorphisms $$F^k: \ \ \pi_1(T^n)\to \pi_1((\mathbb C^*)^n)$$ Is it true that $...
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1answer
28 views

Fundamental group of the quotient space of the $\Bbb Z$-action on $X=\Bbb C-\{0\}$ given by $n\cdot z=\lambda^nz$

Let $\lambda$ be a fixed nonzero complex number whose modulus is not $1$. Consider the $\Bbb Z$-action on $X=\Bbb C-\{0\}$ given by $n\cdot z=\lambda^nz$. I want to compute the fundamental group of ...
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1answer
84 views

Prove that $G \cong \pi_{1} (X/G)$.

This is a question from an exam. Let $X$ be a topological space which is simply-connected, and let $G$ be a group of homeomorphisms of $X$ which acts properly discontinuously, meaning $$\forall \ x\in ...
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1answer
89 views

Compute the fundamental group of $\mathbb{C}^*/\Gamma$, where $\Gamma=\{\varphi^n:\varphi(z)=4^nz,n\in\mathbb{Z}\}$

Problem Find the fundamental group of the orbit space $\mathbb{C}^*/\Gamma$, where $\mathbb{C}^*=\mathbb{C}\backslash\{0\}$, and $\Gamma=\{\varphi^n:\varphi(z)=4^nz, n\in\mathbb{Z}\}$ acts on $\mathbb{...
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1answer
49 views

Fundamental groups of coset spaces

Let $G,H$ be topological groups. In the case that $G$ is connected and simply connected, we have $$\pi_1\left(G/H\right)\simeq\pi_0\left(H\right)\simeq H/H_0$$ where $H_0\leq H$ is the component of $H$...
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1answer
68 views

Computing the homotopy type of the total space of a fiber bundle

This is an example/exercise from section 13 in Differential Forms in Algebraic Topology by Bott & Tu. I've been posting a few questions to get through this part of the book. It was pretty tough ...
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2answers
51 views

fundamental group of $S^{1}\cup\left(\mathbb{R}\times0\right)$ (Theta space)

I was asked to find the fundamental group of $S^{1}\cup\left(\mathbb{R}\times0\right)$. I was given three options - $\mathbb{Z}$, the fundamental group of two spheres connected by one point (i.e F2 - ...
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1answer
23 views

Induced homomorphism between fundamental groups of annulus

I have the following annulus in $\mathbb{C}$: $U = \left\{z \in \mathbb{C} : 0 < |z|<1 \right\}$ and $V = \left\{z \in \mathbb{C} : 2 < |z|<3 \right\}$ And I have to prove that the induced ...
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1answer
46 views

What is meant by “covering space corresponding to a subgroup of fundamental group”?

I have the following situation: $K$ is a finite simplicial complex and $\widetilde{K}\to K$ is "a covering space corresponding to the subgroup $H$ of $G=\pi_1(K)$". Can anyone say what ...
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1answer
59 views

Uncountable fundamental group.

I was trying to understand what is an example of a topological space that has an uncountable fundamental group. I was reading this answer but I don't understand what $L_q \equiv 0$ and $L_q \equiv 1$ ...
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0answers
45 views

Show that if the induced homomorphism $f_*:\pi_1(T\#T,x_0) \to \pi_1(T\#T,f(x_0))$ is 1-1 then it is onto.

Let $T$ be a torus and suppose $f: T\#T \to T\#T$ is a continuous map. Show that if the induced homomorphism $f_*:\pi_1(T\#T,x_0) \to \pi_1(T\#T,f(x_0))$ is 1-1 then it is onto. I know that $ \pi_1(T\#...
2
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1answer
63 views

Calculate $p_*(\pi_1(\tilde{X},e_i))$

Give an example of a two-fold cover $(\tilde{X},p)$ of figure eight. For those examples choose a basepoint $e$ and a base point $e_i\in \tilde{X}$ and calculate $p_*(\pi_1(\tilde{X},e_i))$ My attempt: ...
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1answer
24 views

Connected components of free loop space

Let $X$ be a topological space. And let $\Lambda X=\mathrm{Top}\left[S^1,X\right]$ be the space of continuous loops in $X$. Then how do we calculate $\Pi_0\Lambda X$, the set of connected components ...
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2answers
29 views

For a topological manifold $X$ is it true that $X$ is a covering of $X\lor X$?

Here is my question: Let $X$ be a topological manifold. Is it true that $X$ is a covering of $X\lor X$ and $X\lor X\lor X$ and, so on. I have a intuition, $\pi_1(X\lor X)=\pi_1(X)*\pi_1(X)$.
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42 views

How lemma 2 is a corollary of lemma 1?

We took this lemma 2: If there is a pointed homotopy between $f$ and $g$ where they are functions from $(X, x_{0})$ to $(Y, y_{0})$ then $$f_{*} = g_{*}.$$ And our professor said (without proof) that ...
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0answers
53 views

Lifting of homotopy

Let $F: I \times I \rightarrow \mathbb{S}^1$ a continuous function such that $F(0,t)=F(1,t)=p \in \mathbb{S}^1$ for a fixed $p$ (i.e. a homotopy between two loops in the circle based at $p$). Show ...
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1answer
55 views

A quick question regarding notation and translating it to spoken language in Munkres

Let $h : X \rightarrow Y$ be continuous, with $h(x_0) = y_0$ and $h(x_1) = y_1$. Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$, and let $\beta= h \circ \alpha$. Show that $$\hat{\beta} \circ (h_{...
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1answer
38 views

Covering mappings

If $X$ is path conneceted, fundamental group of $X$ is finite and there is a covering map $p: X \to S^1$ then $X$ is simply conneceted (1 connected) Any ideas how to prove this?
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0answers
63 views

Fundamental group of the complement of a divisor

Let $X$ be a smooth compact projective manifold of dimension $n \geq 2$ which is simply connected. Take $D \subset X$ be a smooth divisor which is isomorphic to the disjoint union of $s>1$ copies ...
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1answer
38 views

A compact hyperbolic manifold and fundmental group [closed]

I just start to learn differential geometry and have a problem. Let $M$ be a compact hyperbolic manifold, how to prove Z $\oplus$ Z is not a subgroup of $\pi_1(M)$ .
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1answer
29 views

Inclusion of a subspace $X$ into $RP^n$ Induces Surjection on $\pi_1$

Studying for a qualifying exam, and came across this problem: Let $p: S^n \to RP^n (n \geq 2)$ be the standard two-fold covering, and let $X \subset RP^n$. Prove that $p^{-1}(X)$ is path-connected if ...
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1answer
34 views

Prove that 1-form is exact using the fundamental Group Homomorphism

I have the following group homomorphism: $\pi_1(i,z_o):\pi_1(U,z_o)\rightarrow \pi_1(\mathbb{C},z_o)$ where $i$ is the inclusion map and I know that it is the null-homomorphism, i.e. $\pi_1(i,z_o)([\...
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1answer
50 views

Fundamental group of $X$, a CW complex is isomorphic to the fundamental group of its 2-skeleton

I'm trying to show that if $X$ is a CW-complex, then $$ \pi_1(X) = \pi_1(X^2)$$ where $X^2$ is the 2-skeleton. I found the following proposition in Hatcher's book: Proposition 1.26. (a) If $Y$...
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1answer
20 views

Proving map si covering map of torus

Let $f:\mathbb{R^2} \rightarrow \mathbb{T^2}$ be defined as $$f(x,y)=(x-[x],y-[y]),$$ where $[x]$ denotes the integer part of $x$. I would like to show that this is a covering map of the torus $\...
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0answers
37 views

Van Kampen Theorem when $A \cap B$ is a circle bounding discs from $A$ and $B$.

We are given a finite path-connected simplicial complex $X = A \cup B$, where $A$, $B$ and $A \cap B$ are sub-simplical complexes. If $A \cap B$ is homeomorphic to a circle and bounds discs in each ...
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1answer
39 views

The fundamental group of an inverse limit of an inverse system of topological spaces

If I have an inverse system of topological spaces, which fundamental group is $\mathbb{Z}$, would the fundamental group of its inverse limit be $\mathbb{Z}$? And if not, under what conditions is it ...
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1answer
62 views

Fundamental Group of a free G-space

It is very well known that any group $H$ can be the fundamental group of a topological space. What happened if we restrict the class of topological spaces to the free equivariant topological spaces? ...
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1answer
44 views

How to prove that the n-th root is not continuous using the Fundamental Group

I need to prove that the function $g:\mathbb{C}^∗ \rightarrow \mathbb{C}^*$ such that $(g(z))^n=z$ is not continuous using the fundamental group. I tried to use the argument in the question: How to ...
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1answer
73 views

Fundamental group of the complement in $\Bbb R^3$ of the union of the $x$-axis, the $y$-axis, and the cylinder $S^1\times [0,1]$

Let $A$ denote the union of $x$ and $y$ axes in $\Bbb R^3$, and let $B$ denote the cylinder $S^1\times [0,1]$ in $\Bbb R^3$. I am asked to compute the fundamental group of the space $X=\Bbb R^3-(A\cup ...
5
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1answer
167 views

Finding subgroups of $G=\langle x,y,z~|~x^2y^2z^2 \rangle$ using covering space theory

Consider the group $G$ having a presentation $G=\langle x,y,z~|~x^2y^2z^2 \rangle$. I am trying to find all subgroups of $G$ of index 6 using covering space theory. It is well-known that the connected ...
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1answer
69 views

How to prove that the complex logarithm is not continuous using the Fundamental Group

I need to prove that the function $f:\mathbb{C^*} \rightarrow \mathbb{C} ; \exp{(f(z))} = z$ is not continuous using the fundamental group. I´ve found this Does every continuous map induce a ...
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1answer
58 views

Seifert Van Kampen Example

I'm stuck in the explicit calculation of the fundamental group of one space. I have the space that is three copies of $\mathbb{S}^{1}$ disposed in vertical, for example, $L=\partial(B[(0,0),1])\cup\...
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0answers
25 views

Čech cocycles and monodromy

It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
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1answer
65 views

Let $S$ be containing $n$ distinct points of $\mathbb{R}^{2}$, then prove $\mathbb{R}^{2}-S \simeq \vee_{i=1}^{n}S^{1}$

In order to solve the fundamental group of $\mathbb{R}^{2}$ minus $n$ distinct points i stumbled across the following problem : I'd have liked to create the following homopoty equivalence between $S^...
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1answer
42 views

Use Seifert-van Kampen to compute the fundamental group

I am trying to generalize the problem I ask yesterday Fundamental group of sphere with antipodal points on the equator, i.e. the question is "Compute the fundamental group of the space obtained from ...
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2answers
40 views

No closed orientable 3-manifold is homotopy equivalent to $S_g\vee S^3$

I am working on ALgebraic Topology in my past Quals. "Prove that no closed orientable 3-manifold is homotopy equivalent to $S_g\vee S^3$, where $S_g$ is the orientable surface of genus $g\...

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