For questions about or involving covering spaces.

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4
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109 views

An entire function with finite covering group is a polynomial.

Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
0
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2answers
180 views

Lifting an automorphism to the universal covering space..

Let $X$ be a manifold and $Y$ be its universal covering. Is it true that any $\phi \in \mathrm{Aut}(X)$ can be lifted to $\overline{\phi}\in \mathrm{Aut}(Y)$?
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0answers
35 views

Classification theorem of the coverings of a given space

I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as Classification of covering spaces ...
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0answers
50 views

How to find every 4-sheet covering of the wedge sum?

Based on this question 4-sheet covering of the wedge sum of two circles I know how to find one 4-sheet covering of the wedge sum, but how to find every 4-sheet covering of the wedge sum? I really ...
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0answers
92 views

Covering space (Lie groups and their maximal tori)

Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
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0answers
32 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
2
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0answers
59 views

Global sections of covering spaces

Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$. Is there any reference where this is explicitly ...
4
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2answers
486 views

Covering space of a non-orientable surface

I have the following problem: Find the 2-sheeted (orientable) cover of the non-orientable surface of genus g. The cases $g=1,2$ are well-known, we have that the cover of $\mathbb{R}P^2$ is ...
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1answer
147 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
113 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 ...
4
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1answer
121 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
2
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1answer
68 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
59 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 $\pi$ ...
4
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1answer
78 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 ...
0
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1answer
442 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. ...
2
votes
2answers
134 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 $f(0)=x_0$ ...
3
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4answers
833 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 ...
4
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2answers
273 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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0answers
62 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 ...
2
votes
1answer
292 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
264 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 ...
0
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1answer
74 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 ...
2
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0answers
91 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 ...
4
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1answer
564 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
223 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 ...
0
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1answer
255 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 ...
4
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1answer
154 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 ...
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1answer
249 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 ...
2
votes
1answer
766 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. ...
0
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1answer
207 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
104 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 ...
4
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2answers
225 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
282 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
217 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
203 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 ...
2
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0answers
103 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 ...
5
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1answer
429 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
353 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 ...
2
votes
3answers
386 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
573 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
408 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
60 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 ...
3
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1answer
86 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
63 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
231 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
451 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 ...
2
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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 ...
4
votes
1answer
79 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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votes
3answers
502 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 ...
3
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1answer
388 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 ...