For questions about or involving covering spaces.

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3
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1answer
47 views

Universal covering space of connected open subset of $\mathbb R^n$

Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?
3
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2answers
157 views

covering space of a particular CW complex

I am trying to find all connected covers of the following space $X$ (up to isomorphisms) $X$ has one $0$-cell, two $1$-cells labeled $a$ and $b$, and three $2$-cells attached via $a^2$, $b^2$ and ...
1
vote
2answers
48 views

Show that f is covering map and find covering tranformation group

Prove that $f:\mathbb{R}^2\to T^2$ defined by $f(x,y)=(e^{2\pi i x},e^{2\pi iy})$ is covering map and also find covering tranformation group$=\{g:\mathbb{R}^2\to\mathbb{R}^2\mid g$ is diffeomorphism ...
0
votes
1answer
46 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
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0answers
54 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
1
vote
1answer
36 views

What subsets of a covering space cover their image?

Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map? If it makes things easier, assume $E$ is simply connected, that is, the ...
1
vote
1answer
47 views

Finite group acting freely on Haussdorf space- Topology problem

How to prove the following problem: It is given Hausdorff space $X$ and finite group $G$ (with neutral $e$) that is acting freely on $X$. For $g\in G$, $\overline{g}:X\rightarrow X$ is ...
0
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2answers
114 views

How to find the induced map $f_{*} : \pi_1 (S^1 , (1,0)) \to \pi_1 (S^1 , (1,0) ) \ \ ? $

I came across this old exam question while studying for my own exam for our topology course. Let $f : S^1 \to S^1 $ be the map $z \mapsto z^n$. What is the induced map $$f_{*} : \pi_1 (S^1 , (1,0)) ...
3
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0answers
299 views

Algorithms for covering a rectilinear polygon using rectangles of the same size

The following is the problem description: All angles of the polygon are right. It may be convex or concave. Use rectangles of the same size to cover the polygon. The edge of the polygon and ...
0
votes
1answer
65 views

function lifting on $S^1 \times S^1$

Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 ...
2
votes
1answer
61 views

Spin group without Clifford algebras

I have to build the spin group $Spin(n)$ without use Clifford algebras. Can I find a complete description of spin group with a topological method? How can I build $Spin(n)$ as the double covering of ...
1
vote
1answer
50 views

Finding sheet number of torus using universal cover

I have a question from my lecture notes that I need clearing up: Given a covering $p: S^1\times S^1 \rightarrow S^1\times S^1$ by $p(z,w)=(z^a w^b,z^c w^d), a,b,c,d\in\mathbb{Z}$, we want to find the ...
5
votes
2answers
381 views

Covering space Hausdorff implies base space Hausdorff

There is an exercise problem in Hatcher's Algebraic Topology book asking to show that if $p:\tilde{X}\rightarrow X$ is a covering space with $p^{-1}(x)$ finite and nonempty for all $x\in X$, then ...
1
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0answers
35 views

The covering space of a region contained in complex plane delete two points.

We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
3
votes
1answer
178 views

The restriction of a covering map on the connected component of its definition domain

Suppose $p:Y\to X$ is a covering map, $X,Y$ are manifolds and $X$ is connected. If $Z$ is a connected component of $Y$, I wonder if the restriction of $p$ on $Z$ is also a covering map? If not, what ...
2
votes
2answers
131 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
5
votes
1answer
166 views

Induced map on homology from a covering space isomorphism

Suppose $S^1 \times \mathbb{R}P^2$ covers some space. Why is it that any covering space isomorphism $h$ induces the identity map on $H_1$? I don't see how to prove this except maybe from looking at ...
3
votes
1answer
74 views

Is $f$ necessarily a covering?

Let $f : X \rightarrow Y$ be a continuous map of spaces where $X$ is compact Hausdorff , $Y$ is Hausdorff and both spaces are path-connected and locally path-connected. Suppose that for every $x \in ...
4
votes
1answer
77 views

Restriction of Covering Space

I'm studying for an exam, and got stuck on the following exercise: Find all two-sheeted covering spaces for $X =\mathbb{S}^1 \vee \mathbb{S}^1$. Label the two circles of $X$ by $a$ and $b$. Attach ...
0
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0answers
45 views

Which of the following spaces nontrivially cover themselves?

I am having some difficulties with a qualifying exam question. I would appreciate if someone could give me a little help. Which of the following spaces nontrivially cover themselves? (a) $S^3$ (B) ...
5
votes
1answer
226 views

The action of the group of deck transformation on the higher homotopy groups

This is for homework. I'm supposed to do exercise 4.1.4 in Hatchers "Algebraic Topology", which is to show that given a universal covering $p: \tilde{X} \to X$ of a path-connected space $X$, the ...
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0answers
53 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
2
votes
1answer
81 views

All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
7
votes
2answers
294 views

A locally constant sheaf on a locally connected space is a covering space; Proof?

As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
1
vote
1answer
59 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
0
votes
1answer
50 views

Plane rotation homomorphism

I want to show that the following set (given group structure with matrix multiplication) $$D=\left\{ D(\theta)=\begin{pmatrix} e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2} \end{pmatrix}, 0 \leq ...
2
votes
3answers
122 views

Relationship between the fundamental group and the natural equivalence classes of its universal cover

For a universal covering $p: Y \to X$, under the equivalence relation $y_1 \sim y_2$ if $p(y_1) = p(y_2)$, $Y$ admits the quotient map $\, \, \, q: Y \to Y / \sim$. There is a natural bijection $\bar ...
2
votes
1answer
50 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
4
votes
2answers
114 views

Extending a quotient map to a covering map on $\mathbb{RP}^2$

Why can we not extend the quotient map $q:[0,1]\times[0,1] \to \mathbb{RP}^2$ to a covering map, $\mathbb{R}^2 \to \mathbb{RP}^2$?
2
votes
2answers
219 views

Universal cover of a figure eight?

An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'. Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can ...
2
votes
1answer
84 views

Universal cover modulo the monodromy action

Let $\tilde{X} \to X$ be the universal cover of a connected, locally path-connected and semi-locally simply connected topological space $X$. Is it always true that the orbit space $$ \tilde{X} \;/\; ...
2
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2answers
110 views

Is the universal covering surface orientable?

Let $M$ be a smooth, say also closed (compact and without boundary) surface. Is it true that its universal covering surface is orientable?
3
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1answer
118 views

Domain is Hausdorff if image of covering map is Hausdorff

Suppose that $p:X\rightarrow Y$ is a covering map. Show that if $Y$ is Hausdorff, then so is $X$. I have an answer but I'm not sure if it's right? By definition of Hausdorff, $\forall x,y, \in Y, ...
2
votes
0answers
142 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then ...
2
votes
1answer
234 views

Domain is locally path-connected if image of a covering map is locally path-connected.

Show that if $p:X\rightarrow Y$ is a covering map and $Y$ is locally path-connected, then so is $X$. How do you go about proving this? I can think of two ways of doing this, either by definition of ...
3
votes
2answers
172 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
0
votes
1answer
36 views

Defining a homeomorphism from an equivalence relation on a covering map domain to its image.

If $p: \tilde Y \to Y$ is a covering map, and we define an equivalence relation $\sim$ on $\tilde Y$ by $\tilde y_1 \sim \tilde y_2$ if $P(\tilde y_1) = p(\tilde y_2)$. How would you show that the ...
0
votes
1answer
70 views

Showing the fibre over a point in a covering map is a discrete space.

If $ p : \tilde Y \to Y$ is a covering map, how would you show that for every $y \in Y$ we have that $p^{-1}(y)$ is a discrete space?
2
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0answers
51 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
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0answers
67 views

Question on homotopy lifting

I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear. A lemma in my lecture notes says: Let $p: \tilde Y \to ...
3
votes
2answers
158 views

Why must a finite covering map be closed?

A covering map $p:C\to X$ is finite when for each $x\in X$ we have $|p^{-1}(x)|<\infty.$ I have to prove that such a covering map has to be closed. I'm having trouble with it. When $p$ is a ...
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0answers
98 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
5
votes
1answer
92 views

Classifying a Branched Covering Space

This question comes from the proof of proposition 2.2 in Henry Laufer's 'Normal Two-Dimensional Singularities" text. I am excerpting the part I don't understand, and I think it's a self-contained ...
3
votes
2answers
79 views

Is this covering map a homeomorphism?

Suppose $\pi : \widetilde X \to X$ is a finite, connected covering, and suppose that there exists a continuous map $f: \widetilde X \to \mathbf R^2$ which is injective on each fibre of $\pi$. Is $\pi$ ...
0
votes
2answers
129 views

How to prove that $p:S^1\rightarrow S^1$ $z\mapsto z^2$ is a covering map?

How can I prove that $p:S^1\rightarrow S^1$, $z\mapsto z^2$ is a covering map? Please help. I was not able to prove it by applying definition of covering space.
13
votes
1answer
347 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
4
votes
3answers
215 views

proving that a covering map with certain domain and range is homeomorphism

Let $p:E\to B$ be a covring map, with $E$ path connected. Show that if B is simply connected, then $p$ is a homeomorphism. Well I don't know exactly what can I do here, maybe I have to start with ...
4
votes
1answer
229 views

Covering space homeomorphism

In the course of an exercise from Hatcher's topology text, I came to the following point. Given $p: \tilde{X} \to X$ the universal cover for $X$, and a continuous map $h: \tilde{X} \to \tilde{X}$ ...
3
votes
1answer
334 views

covering map with finite fibres and preimage of a compact set

Let $f:X\to Y$ be a covering map (covering maps are surjective) , Y be compact set. And suppose that $f^{-1}(y) $ is finite for each $y\in Y$. Prove that $X$ is also compact. I think that this ...
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votes
0answers
34 views

Pullback of a family of curves via a covering map.

Let $X$ be a smooth compact projective manifold and $\pi:Y\rightarrow X$ a Galois covering map. Is it always possible to pull back a family of curves on $X$ to $Y$?