For questions about or involving covering spaces.

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154 views

The definition of normal covering in Hatcher book

In page 70 of Hatcher's book, in the section Deck Transformations and Group Actions, the author defines a normal covering in the following way: A covering space $p:\tilde X\to X$ is called normal ...
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1answer
121 views

How can I find the deck transformations of $p:\mathbb R\to S^1$?

How can I find the deck transformations of $p:\mathbb R\to S^1$, where $p(t)=e^{2\pi ti}$? I tried in this way: Let $\phi:\mathbb R\to \mathbb R$ be a deck transformation, so $p(\phi(t))=p(t)$ for ...
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1answer
129 views

What's the automorphism group of this covering?

What's the automorphism group of this covering? I know why this is a covering, but I don't know how to find the automorphism group of this covering. I need help, thanks
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1answer
72 views

Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
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3answers
61 views

Is every fiber-preserving map between coverings again a covering?

Suppose we have two coverings $$p_1:Y\rightarrow X$$ and $$p_2:Y^\prime\rightarrow X$$ and further a continuous map $$\pi\colon Y\rightarrow Y^\prime,$$ such that $$p_2\circ\pi=p_1.$$ Ist $\pi$ ...
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1answer
82 views

Why is the rank of $f_\ast L$ the degree of $f$

Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$? Here is my ...
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1answer
528 views

$4$-sheet covering of the wedge sum of two circles

I'm trying to find the $4$-sheet covering of the wedge sum of two circles I don't know even how to begin, I know just the definitions of coverings and simple examples, I really need help here. ...
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2answers
150 views

Does every lift of a constant path is constant?

I'm trying to prove that every lift of a constant path is constant using the path lifting property which says that for each path $f:I\to X$ and each lift $\tilde x_0$ of the starting point $f(0)=x_0$ ...
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4answers
1k views

Why this map is a covering map?

I'm trying to find the universal covering space of the Klein bottle. I know that $\mathbb R^2$ covers the Klein bottle , but I don't know how to prove, I found this proof on internet: Someone knows ...
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2answers
320 views

About covering maps and sections!

If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?
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66 views

How do I prove that a map is not a covering map?

I'm thinking how to prove that a map is not a covering map. For example let $p:\mathbb R_+\to S^1$ be a map defined by $p(\theta)=(\cos(2\pi\theta),\sin(2\pi\theta))$. I'm trying to find a point which ...
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1answer
383 views

This quotient map is a covering map

For any integer $n\ge 1$ the map $q:\mathbb S^n\to\mathbb {RP^n}$, which identifies antipodal points, is a covering map. I'm trying to solve this question in the following manner (with the help of ...
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1answer
351 views

this map $p(z)=z^n$ is a covering map?

For any positive integer $n$ the map $p:S^1\to S^1$, defined by $p(z)=z^n$ is a covering map. I think it's easy, but as I'm a really beginner I'm struggling to prove it. By the way, what kind of ...
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1answer
97 views

This projection is a covering map?

If $\tilde X$ is the product of $X$ with a discrete space, the projection $\tilde X \to X$ is a covering map. This question seems really easy, but as I'm a beginner there are some things a little bit ...
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0answers
97 views

Are there infinitely many rational functions of bounded degree and given ramification

It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus is finite (up to isomorphism). Edit. The branch locus $B$ of $f:X\to ...
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1answer
678 views

A good way to understand Galois covering?

A covering map $f:X\rightarrow Y$ is called Galois if for each $y\in Y$ and each pair of lifts $x, x^{'}$, there is a covering transformation taking $x$ to $x^{'}$. What is a good way to understand ...
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0answers
255 views

Completelly cover area with minimum number of maxed circles NP-completeness (or harder) proof

everyone. I'm looking for paper with proof of NP-completeness following, or similar problem. Given: Area $S \subset \mathbb{N}^2$, let it be convex or rectangular, I believe it doesn't matter ...
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1answer
300 views

Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$

I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...
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1answer
158 views

Constructing Riemann surfaces using the covering spaces

In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow: A polynomail-like map ...
2
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1answer
326 views

Universal cover via paths vs. ad hoc constructions

I'm looking for some intuition regarding universal covers of topological spaces. $\textbf{Setup:}$ For a topological space $X$ with sufficient adjectives we can construct a/the simply connected ...
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1answer
942 views

What is the optimal solution for covering a rectangle with circles?

Given a rectangle of area n*m, and identical circles with radiuses r. What is the optimal solution for covering this rectangle with minimum number of circles? I found a relative solution here. ...
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2answers
239 views

Covering space and Fundamental group

Let $p:E\to X$ be a covering space and $\pi_1(E)$ be a fundamental group of $E$. Can you give me a recept for calculating a fundamental group $\pi_1(X)$ (may be for some special cases)? Thanks a lot! ...
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0answers
108 views

Universal cover as a principal $\pi_1$ bundle.

Let $M$ be a connected manifold with universal cover $\tilde M$ and fix $x_0 \in M$. Then it is well-known that $\tilde M \to M$ is a principal $\pi_1(M,x_0)$ bundle. I'm a bit confused about the ...
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2answers
257 views

Is a polynomial also a covering map?

Question: Let $p(z)$ be a polynomial over $\mathbb{C}$. Is it true that $p:\mathbb{C} \to \mathbb{C}$ is a covering map ? Partial answer: Let us look first at the points where $p'(z)\ne 0$. There the ...
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1answer
325 views

Universal covering of a connected sum

Maybe it's an easy question: How can we find the universal covering of the connected sum of tori or projective planes? Is there a general method?
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1answer
228 views

Pullback of differential form on the double covering

On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map ...
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1answer
228 views

Covering spaces as quotients of the universal covering

Let $(\tilde{X},p)$ be a universal covering space of $X$. We know that if $G$ acts properly disctinously on $\tilde{X}$, then $\tilde{X}$ is a covering space of $\tilde{X}/G$ and ...
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0answers
106 views

Identifying the numbers of degree $n$ covering spaces of $X$

Let $X$ be a path-connected, locally path-connected and semilocally simply-connected space. Can we find a correspondence between degree $n$ covering spaces of $X$ and group homomorphism ...
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1answer
519 views

Lifting of maps to a covering space

I am reading Algebraic topology by W. Massey and I have a problem with the proof of property 5.1: Let $(\tilde{X},p)$ be a covering space of $X$, $Y$ a connected and arcwise connected space, ...
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1answer
368 views

The covering space of connected space

Let $X$ be a connected topological space, and $\pi : Y \rightarrow X$ a surjective covering space map. Suppose that the group of deck transformations of $\pi$ contains a subgroup $\mathbb Z_p$, where ...
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3answers
433 views

Cayley complex as universal covering space

In Combinatorial Group Theory, Lyndon and Schupp construct a complex $K(X;R)$ from a presentation of group $G=(X;R)$, such that $G \simeq \pi_1(K,v)$ (proposition 2.3, p.117). Moreover, the Cayley ...
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1answer
709 views

How to classify 3-sheeted covering space for $S_{1}\vee S_{1}$?

This might be a duplicate. This question also feels routine (it is also the execrise 10, page 88 in Hatcher). From Harvard qualification exam, 1990. Let $X$ be figure eight. 1) How many 3-sheeted, ...
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1answer
473 views

Universal Cover of projective plane glued to Möbius strip

Consider the usual cell structure on $\mathbb R P^2$, with one 1-cell and one 2-cell attached via a map of degree 2. Consider the space $X$ obtained by gluing a Möbius band along the 1-cell via a ...
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1answer
63 views

automorphisms of varieties with respect to a cover

Let $X$ and $Y$ be (smooth projective connected) varieties over $\mathbf{C}$. Let $\pi:X\to Y$ be a finite surjective flat morphism. Does this induce (by base change) a map $\mathrm{Aut}(Y) \to ...
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1answer
88 views

Why is this covering map doubly periodic?

The universal cover of the torus $T$ is the complex plane $\mathbb{C}$. If $p: \mathbb{C} \to T$ is the covering map, why is $p$ doubly periodic?
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1answer
65 views

Using coverings of graphs

How can I use coverings of graphs to show that if $G$ is a finitely generated free group and $H$ is a subgroup of finite index, then $H$ is finitely generated. I've seen this done without using ...
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0answers
258 views

Monodromy Theorem and Homotopy Lifting Theorem

I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct? Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a ...
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2answers
526 views

Is a covering space of a manifold always a manifold

Assume $M$ is a manifold and $q : E \to M$ is a covering map. I have been told a few times that a covering space of a manifold is again a manifold. Indeed, it is easy to verify that $E$ is both ...
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0answers
71 views

Liftings of curves $u\cdot v$ and $v\cdot u$ with respect to the sine covering map.

I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache: Let $X=\mathbb{C}\setminus\{\pm1\}$ and $Y ...
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1answer
83 views

The universal cover of the multiplicative group over the field of algebraic numbers

Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ ...
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3answers
572 views

deck transformations of the universal cover

One approach to classifying the coverings of a nice space $X$ without choosing a basepoint is to look at actions of the fundamental groupoid on sets. Another way that seems natural to me is to fix a ...
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1answer
446 views

covering spaces and the fundamental groupoid

Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between ...
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1answer
142 views

Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?

I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following: If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of ...
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1answer
211 views

Branch points of rational functions

Let $f$ be a rational function on a compact connected Riemann surface $X$. The rational function $f$ induces a holomorphic map $\overline{f}:X\to \mathbf{P}^1(\mathbf{C})$. Let $x$ be a point on the ...
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114 views

Very Basic Covering Space Q. - Two different covering maps over a same space?

Here's the situation I'm in. I have a map from the (closure of) Upper Half Plane ($\mathbb{U}$) into the punctured (closed) disk ($\mathbb{D}$) called $q$ that satisfies $q(0) = 1$, and $q(z+1) = ...
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0answers
57 views

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...
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78 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
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2answers
253 views

Universal covering of $SO(3,\mathbb{R})$

How do you prove that the universal covering of $SO(3, \mathbb{R})$ is $S^3$ ? Or equivalently, that it is diffeomorphic to $P_3\mathbb{R}$ ? Thank you for your answers.
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2answers
236 views

Are fundamental groups of Riemann surfaces always finitely generated

For any finite subset $B\subset \mathbf{P}^1$, the fundamental group of the Riemann surface $\mathbf{P}^1-B$ is finitely generated. Is this true if we replace $\mathbf P^1 $ by a higher genus compact ...
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1answer
125 views

Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$. Are the following two conditions equivalent? The function field extension $K(Y)\subset ...