Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

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1answer
44 views

What does n times a generator of a cyclic group mean?

It seems to be saying that, for some integer deg(f) it sends the generator i to deg(f) times the generator, but I can only see it making sense if it sends it to the generator^deg(f) What does it mean ...
3
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1answer
153 views

Deck transformation group isomorphic to $\mathbb{Z}/\mathbb{6Z}$

I'm looking for a topological space $X$ and a covering space $(\tilde X,p)$ such that the deck transformation group Aut$_{X}(\tilde X)$ is isomorphic to $\mathbb{Z}/\mathbb{6Z}$. My idea is to find ...
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1answer
56 views

Why is $N(p_*(\pi_1(T, \overline{x}_0)))/p_*\pi_1(X, x_0) \cong D$?

The following is from "Homotopical Topology" by Fuchs et al. Let $p:T \to X$ be a covering, $\overline{x}_0 \in T$, $p(\overline{x}_0) = x_0 \in X$, $D$ the group of deck transformations of $T$, and ...
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2answers
58 views

How to build homotopy from convex to a point?

How to build homotopy from convex to a point? for example I want to build homotopy from the real line $\Bbb R $ to a single point $x_0$ I know that every path in a closed set $[a,x_0]$ in $\Bbb R$ ...
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1answer
59 views

Equivalence of Categories between the Fundamental Group and Groupoid

Let $X$ be a path connected space, and let $x \in X$. Then we have that $\pi_1 (X,x)$ is a full subcategory of $\Pi(X)$. So, the inclusion functor $J: \pi_1(X,x) \to \Pi(X)$ is an equivalence of ...
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0answers
118 views

Covering spaces of $\mathbb{T} \times \mathbb{RP}^2$

Let $X=\mathbb{T} \times \mathbb{RP}^2$. I want to find all covering scpaces (up to isomorphism) of $X$. First, suppose we know all covering spaces of toro $\mathbb{T}$ and projective plane $\mathbb{...
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1answer
91 views

Homology group of $\mathbb{S}^1 \vee \mathbb{RP}^2$ and covering spaces

Let $H_1(X,\mathbb{Z})$ the first homology group of $X$ with integral coefficients, where $X$ is a topological space. In particular I consider $X=\mathbb{S}^1 \vee \mathbb{RP}^2$. Let $x_0$ be the ...
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0answers
66 views

Classification of principal bundles and of regular coverings

Let $G$ be a discrete group and let $X$ be a good space with fundamental group $\pi_1$. We know the following things: The connected principal $G$-bundles over $X$ are exactly the regular coverings of ...
2
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4answers
174 views

Covering spaces of $\mathbb{C}^{\times}$

Let $\mathbb{C}^{\times}=\mathbb{C} \setminus \{0\}$. I'm trying to find out all covering spaces of this space. Let's start. (I'm using Massey's book: Algebraic Topology, an introduction) First of ...
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1answer
186 views

If $A$ is a deformation retract of $X$ and $B$ is a deformation retract of $A$ then $ B$ is a deforemation retract of X

If $A$ is a deformation retract of $X$ and $B$ is a deformation retract of $A$ then $ B$ is a deformation retract of $X$. I am a beginner in Algebraic Topology so I tried to write every proof out ...
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0answers
31 views

Continuity of a lifting $\hat{\varphi}$

I have a problem with the proof of the following proposition: Let $(\hat{X},p)$ be a covering space of $X$, Y a connected and locally arcwise-connected space, $y_0\in Y$, $\hat{x}_0\in X$, and $x_0=p(...
1
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1answer
58 views

Find fundamental group of space $X=(S^1\times S^1)/(S^1\times \{s_0\})$

Find fundamental group of space $X=(S^1\times S^1)/(S^1\times \{s_0\})$ I am not sure how to do it. It seems obvious that $\pi_1(X)=\mathbb{Z}$ because $X$ is torus where one class of nontrivial ...
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2answers
122 views

Prove that $f:X\to S^1$ is null-homotopic iff homomorphism $f_*:\pi_1(X,x)\to\pi_1(S^1,f(x))$ is trivial for some $x$.

Let $X$ be connected and locally path connected space. Prove that: $f:X\to S^1$ is null-homotopic iff homomorphism $f_*:\pi_1(X,x)\to\pi_1(S^1,f(x))$ is trivial for some $x$. I do not have any idea ...
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1answer
82 views

Fundamental group of $\mathbb{R}^3 \setminus \{ \mathbb{S}^1 \text{ and two lines} \}$

Let $\quad r_1 = \{ (x,y,z) \in \mathbb{R}^3 \mid x=0,z=0 \}$ ( $y$ axis), $\quad r_2 = \{(x,y,z) \in \mathbb{R}^3 \mid x=0,y=0 \}$( $z$ axis) and $\quad \mathbb{S}^1 =\{(x,y,z) \in \mathbb{R}^3 \mid ...
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1answer
66 views

Two points from the same topological space that their fundamental group is not isomorphic.

$X$ is a topological space, I need to give example such that for $x_1,x_0 \in X$ $\pi_1(X,x_0) \not \cong \pi_1(X,x_1)$ I think the example is somehow related to the fact that $X$ is not path ...
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0answers
25 views

Instability of even windings of SO(4)

Representatives of $\pi_1(SO(2))=\mathbb{Z}$ may be given by paths$$\theta\mapsto\left(\begin{array}{lr}\cos(n\theta)&-\sin(n\theta)\\\sin(n\theta)&\cos(n\theta)\end{array}\right).$$However, $\...
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0answers
96 views

Role of path connectedness in fundamental group of product space

Proposition 1.12 from Hatcher's Algebraic Topology book states that \begin{align*} \pi_1(X \times Y) \; \text{is isomorphic to} \; \pi_1(X) \times \pi_1(Y) \; \text{if X and Y are path connected} \; \...
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1answer
86 views

Does the Union of simply connected spaces with with certain conditions,simply connected?

Does the Union of finite simply connected open subspaces of a space, with the condition that for each three of these sub spaces the intersection is also simply connected, also have to be simply ...
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0answers
63 views

prove by induction that $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2$

I have demonstrated by Van-Kampen that $\pi_1(\mathbb{R}P^2) =\langle[ab] | [ab]^2 =1\rangle \cong \mathbb{Z}_2$. Using the next connections: $\mathbb{R}P^n \cong S^n\backslash \{\{x,-x\}: |x|=1\}\...
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2answers
171 views

There is no 3-dimensional manifold with fundamental group isomorphic to direct sum of four copies of $\Bbb Z$

I want to show that there doesn't exist any $3$-dimensional manifold for which the fundamental group is isomorphic to the direct sum $\Bbb Z\oplus \Bbb Z\oplus\Bbb Z\oplus\Bbb Z$ I have calculated the ...
3
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1answer
48 views

Homotoping curves in 3/4-manifolds to lie on embedded tori?

I am wondering if every simple closed curve $c$ in a 3-manifold $Y$ can be homotoped to lie on an embedded torus? Maybe I need to assume that the 3-manifold is orientable? [EDIT: As @JohnHughes ...
2
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1answer
33 views

Natural surjection that maps loops to cycles

Theorem: If $M$ is a compact Riemannian manifold of dimension $n$ and has nonnegative Ricci curvature, then $b_1(M) \le \operatorname{dim}M = n$, where $b_1$ is the first Betti number. Proof: There ...
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1answer
166 views

Is this space $S^3$?

I learned here that if we glue two copies of the solid tori together along their boundary, we get $S^3$. What happens if the gluing map is more complicated? In particular, recall that each $A \in \...
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1answer
170 views

Fundamental group obtained by glueing two mobius bands

Let us glue two Mobius Bands $A$, $B$ together along their boundaries. We want to apply Van-Kampen's theorem to their union. The intersection $L$ is homotopic to $S^1$. So we get $$\pi_1(X) = \mathbb{...
3
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1answer
44 views

Non-Homeomorphicity of Compact surfaces

Using the fact that every compact surface is homeomorphic to either a connected sum of torii or a connected sum of real projective planes, and using the fact that the corresponding fundamental groups ...
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0answers
50 views

triviality of etale fundamental group vs triviality of topological fundamental group

Let $k$ be a number field with a fixed embedding to $ \mathbb{C}$. If $V$ is an algebraic variety over $k$ and $V_\mathbb{C} = V \times_k \mathbb{C}$ has nice enough properties one can show that the ...
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1answer
62 views

Fundamental group of wedge sum of two coni over Hawaiian earrings

Consider a conus $C\mathbb H$ over the Hawaiian earring $\mathbb H$ with the bad point $p\in\mathbb H$. Its fundamental group is trivial. However, I think the fundamental group of the wedge sum $C\...
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1answer
66 views

definition of first homology group

I am using a definition of the first homology group by Miranda (Algebraic Curves and Riemann surfaces), which is as follows, The fundamental group $\pi_1(X,\alpha)$ is the group that consists of ...
5
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3answers
307 views

Homotopy group of three spheres

Suppose we take three spheres $X_1 , X_2 , X_3$ and identify the north pole of $X_i$ with the south pole of $X_{i+1}$, where $i$ is taken mod $3$. I.e. We sort of have a 'triangle' of spheres. I want ...
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1answer
117 views

Fundamental group of space form by gluing boundary of Mobius band to $S^1 \vee S^1$

Let $X$ be the space formed by gluing the boundary of the Mobius band around $S^1\vee S^1$ along the path $a \cdot b$ (see figure.) I would like to use Van Kampen's theorem to find a presentation for ...
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0answers
131 views

Inclusion of path connected component induces isomorphism between fundamental groups.

So, I am presented with an excercise that states: Prove that if $X_0 \subset X$ is the path-connected component of that contains $x_0$, then the inclusion $i: X_0 \to X$ induces an isomorphism between ...
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1answer
122 views

de-Rham cohomology $H^1(S)$ and $ \operatorname{Hom}(\pi_1(S, s_0), (\mathbb{R},+))$ are isomorphic?

Let $S$ be a connected smooth 2-dimensional manifold. Let $H^1(S)$ be its first De-rham cohomology group. \begin{align*}\operatorname{Hom}(\pi_1(S, s_0), (\mathbb{R},+))=\{f \vert f :\pi_1(S, s_0)\to (...
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1answer
66 views

Show that $\pi_1 ( \mathbb{R}^2 - \mathbb{Q}^2)$ is uncountable

This is question 1.2.17 from Hatcher, Algebraic Topology: Show that $\pi_1 ( \mathbb{R}^2 - \mathbb{Q}^2)$ is uncountable My line of thinking is trying to show that there exists an injection $$ \...
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0answers
49 views

Fundamental Group of $GL_r(\mathbb{C})$ [duplicate]

Let $r \ge 2$. How to prove that the fundamental group $\pi_1(GL_r(\mathbb{C}))$ of the invertible matrices $GL_r(\mathbb{C})$ over $\mathbb{C}$ equals $\mathbb{Z}$. What about $\pi_1(GL_r(\mathbb{R})...
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2answers
58 views

Let $p: E\rightarrow B$ and $p':E' \rightarrow B$ covering map, E and E' path-connected and locally path-connected, isomorphismus between E and E'

Let $p: E\rightarrow B $ with $b_0 \in B, e_0\in E$ such that $p(e_0)=b_0$ a covering space. Let $p':E' \rightarrow B$ covering maps, and let $e'_0 \in E' $ such that $p'(e'_o)=b_0$. Let now E and E' ...
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1answer
50 views

Show that if $X$ is homotopically equivalent to $Y$, then $\widetilde{X}$ is homotopically equivalent to $\widetilde{Y}$ [duplicate]

Suppose that $X,Y$ are arc-connected and locally arc-connected spaces and that $p:\widetilde{X}\to X$ and $q:\widetilde{Y}\to Y$ are universal covering of $X$ and $Y$ respectively. Show that if $X$ is ...
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2answers
178 views

Find the universal covering of $\mathbb{R}P^2\vee\mathbb{R}P^2$

Find the universal covering of $\mathbb{R}P^2\vee\mathbb{R}P^2$ I know that $\mathbb{R}P^2\cong \mathbb{S}^2/\sim$ for antipodal action and that $\pi_1(\mathbb{R}P^2)=\mathbb{Z}_2$, with which $\pi_1(...
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0answers
50 views

Classify, except isomorphism, all the associated covering spaces of $3$ sheets on $\mathbb{S}^1\times\mathbb{S}^1$. Do the same for the

Classify, except isomorphism, all the associated covering spaces of $3$ sheets on $\mathbb{S}^1\times\mathbb{S}^1$. Do the same for the covering spaces of $4$ sheets. I know that $G=\mathbb{Z}^2$ ...
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0answers
36 views

Let $K$ be the Klein bottle. Give examples of cover spaces $p_1:E_1\to K, p_1:E_2\to K$ each with two sheets and such that $E_1$ and $E_2$

Let $K$ be the Klein bottle. Give examples of cover spaces $p_1:E_1\to K, p_1:E_2\to K$ each with two sheets and such that $E_1$ and $E_2$ are arc-connected but not isomorphic as covering spaces. I ...
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1answer
58 views

Find the fundamental group of (Z × Z) ∗ Z and (Z ∗ Z) × Z

this is a topology question, it's like this: Find the fundamental group of 1) (Z × Z) ∗ Z. 2) (Z ∗ Z) × Z. 3)Z ∗ · · · ∗ Z where there are n copies Here, Z stands for the integer ...
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1answer
110 views

Find a space whose fundamental group is $\mathbb Z/2 × \mathbb Z$

Find a space whose fundamental group is i) $\mathbb Z/2 × \mathbb Z$ ii) $\mathbb Z/2 ∗ \mathbb Z$ Here, $\mathbb Z$ is the set for integers. And $*$ is the free product defined as $F(G \...
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1answer
151 views

Topological structure of the dunce hat

I'm trying to get an idea on how the dunce hat works. I'd like to compute $\pi_1$, $H_n$, and its universal covering space. Any ideas on what these will look like?
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1answer
125 views

Are spaces $X, Y$ with $\pi_{1}(X) \cong \pi_{1}(Y)$, $\widetilde{X} \simeq \widetilde{Y}$ necessarily homotopy equivalent?

Let $X,Y$ be path-connected, locally path-connected, semilocally simply-connected spaces with isomorphic fundamantal groups and homotopy equivalent universal covers. Are $X$ and $Y$ necessarily ...
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0answers
71 views

Show that there are exactly two spaces obtained from identifying edges of a triangle.

I think I can visualize one of them, which would be identifying two edges together to get a cone. Would the other one be identifying all three edges together for a disk? I have a harder time actually ...
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1answer
66 views

Why is no covering space of $\mathbb{R}P^2 \vee S^1$ homeomorphic to an orientable surface?

I know that $\pi_1(\Sigma_g) = \langle a_1,b_1,\cdots,a_g,b_g | [a_1,b_1]\cdots[a_g,b_g]\rangle$, where $\Sigma_g$ is the orientable surface of genus $g$, and $[a_i,b_i]$ is their commutator. The idea ...
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1answer
142 views

How to define an action of $\pi_1(E)$ on $\pi_n(F)$ for a fibration $F\to E\to B$?

Given a fibration $F\to E\to B$, how to show that the action $\pi_1(F)$ on $\pi_n(F)$ can factor through $\pi_1(E)$: $\pi_1(F)\to\pi_1(E)\to\text{Aut}(\pi_n(F))$? This is exercise 4.3.10 from Hatcher'...
2
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1answer
102 views

how to calculate the fundamental group using van kampen theorem?

i'm really stuck in calculating the fundamental group (pi1 ) of this surface... can someone please help me?.... i'm trying to use the van Kampen theorem but i don't know how to take the open sets here ...
3
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0answers
53 views

If i rotate a shape over a tangent line does this act maintain the fundamental group?

I don't know if this is true or not. If I rotate a shape 360 degrees over a tangent line, does this act maintain the fundamental group ($\pi_1$)? For example, I get a ruined doughnut from $S^1$. I ...
3
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2answers
268 views

Use Van Kampen Theorem to find the fundamental group of a circle $S^1$ joining a sphere $S^2$

this is a topology question: Compute the fundamental group of a Christmas ball, obtained by joining a copy of the circle $S^1$ to a copy of the sphere $S^2$. My thoughts: Intuitively, the ...
2
votes
1answer
178 views

Find a covering space of $T$ corresponding to the subgroup of $\mathbb{Z}\times\mathbb{Z}$ generated by the element $m\times 0$

I know that $p: \mathbb{R}\to \mathbb{S}^1, t\mapsto(\cos2\pi t, \sin2\pi t)$ is a covering space and so $p\times p:\mathbb{R}\times\mathbb{R}\to \mathbb{S}^1\times \mathbb{S}^1$ is also, and since $\...