The covering-spaces tag has no wiki summary.
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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 ...
3
votes
1answer
77 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 ...
1
vote
1answer
113 views
Galois covers of Riemann surfaces
Let $G$ be a finite abelian group, and $C$ a compact Riemann surface (algebraic curve) of genus $g$. I am interested in topological Galois $G$-covers $X \to C$, aka \'etale $G$-principal bundles over ...
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1answer
94 views
Basic covering space question
Given a path connected metric space $X$ and a cover $\tilde{X}$ which is also a path connected metric space with covering map $E$, then is $E$ a local isometry?
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0answers
95 views
What is the Hurwitz number of an elliptic curve
One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points.
Suppose that $X$ is an elliptic ...
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2answers
166 views
Does a morphism between covering spaces define a covering?
My question involves topological spaces $X$, $Y$ and $Z$, two coverings $p : Y \rightarrow X$ and $q: Z \rightarrow X$ of $X$ and a morphism $f: Y \rightarrow Z$ of coverings, i.e. a map which ...
2
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1answer
64 views
What is the length of the following local ring
Let $f:Y\to X$ be a finite etale cover of smooth projective connected varieties. (Or, just a finite degree connected topological cover of connected Riemann surfaces.)
Let $y\in Y$ and let $x=f(y)$. ...
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5answers
575 views
Covering spaces - why are they useful?
As someone who trained as a physicist, I have known for ages that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so, during an idle day at the office I decided to look up what ...
2
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2answers
147 views
metric on the universal covering of a geometric manifold
We know that the universal covering of a closed hyperbolic 3-manifold can be identified with the hyperbolic space $\mathbb{H}^3$. Now, what is not very clear to me is how this identification has to be ...
2
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0answers
107 views
Hyperbolic Universal Covering Space
I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
3
votes
1answer
149 views
discriminant of an étale cover of an elliptic curve
Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1.
Edit: Assume $X$ and $E$ have semi-stable ...
1
vote
1answer
113 views
Number of ramification points in a simple cover
Let $f:X\to \mathbf{P}^1$ be a simple cover of the Riemann sphere. This means that $f$ is a branched cover, and that each fibre has at least $\deg f-1$ points in it.
Is it true that the number of ...
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2answers
211 views
Question about two simple problems on covering spaces
Here are two problems that look trivial, but I could not prove.
i) If $p:E \to B$ and $j:B \to Z$ are covering maps, and $j$ is such that the preimages of points are finite sets, then the composite ...
3
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3answers
612 views
Another Question in Hatcher
First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again.
The problem is 1.3.15. It reads ...
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2answers
109 views
An action of a group on a covering space
We see $S_3$ as the quotient of the free group on two elements and the normal subgroup $R$ generated by $\langle\sigma^3,\tau^2,\sigma\tau\sigma\tau\rangle$ where $\sigma$ and $\tau$ are the ...
2
votes
1answer
201 views
The covering space of a bouquet of 2 circles corresponding to a normal subgroup
Consider $S_3$ with this presentation: $S_3=\left\langle\sigma,\tau:\sigma^2=1, \sigma\tau=\tau^{-1}\sigma\right\rangle$. Let F be the free group with two generators $s,t$ and $R$ the minimal normal ...
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0answers
877 views
The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent
This is exercise 1.3.8 in Hatcher:
Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then ...
2
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1answer
332 views
Composition of coverings of path connected spaces
Do there exist covering maps $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ such that $X$ is path connected and the composition $q\circ p$ fails to be a covering map?
3
votes
1answer
226 views
Irregular covering space of $\mathbb{R}P^2\vee\mathbb{R}P^2$
This was on my final last semester (to find such a cover), and I missed it.
Here are my thoughts on it since then:
I know that the universal cover of $X = \mathbb{R}P^2\vee\mathbb{R}P^2$ is (loosely) ...
7
votes
2answers
413 views
Calculating monodromy
I'm right now learning about Monodromy from self-studying Rick Miranda's fantastic book "Algebraic Curves and Riemann surfaces". Today, I read about monodromy, and the monodromy representation of a ...
1
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1answer
63 views
G-complexes and regular covering
Suppose $X$ a free $G$-complex (i.e. a CW-complex with a free $G$-action that permutes the cells). I would like to show that the projection $$X\overset{p}{\to}X/G$$ is a regular covering spaces with ...
2
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1answer
181 views
Monodromy correspondence
Lately I've been studying monodromy and covering maps (in particular ramified covering mapos of Riemann surfaces), and I came across something I didn't fully understand. Let $V$ be a connected real ...
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0answers
41 views
How to construct certain cover given in Mumford's Abelian Varities book
In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
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2answers
276 views
Why is the Long Line not a covering space for the Circle
I know of several reasons why the long line can't be a covering space for the circle, but I'm more curious in what exactly goes wrong with the following covering map.
Let $L$ be the long line and ...
