For questions about or involving covering spaces in algebraic topology.

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Surface groups and subgroups of fundamental groups

The fundamental group of any closed surface is a surface group. Let $S_3$ be the orientable surface of genus 3. Is $\pi_1(S_3)$ isomorphic to an index-3 subgroup of any surface group? We have 1 ...
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39 views

Covering a rectangle with circles

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1answer
20 views

Covering space of an abelian topological group is abelian if the covering map is a homomorphism

I'm trying to show that if $(E, \cdot)$ and $(G, \cdot)$ are both topological groups, $G$ is abelian, and $(E, p)$ is a covering of $G$ such that $p:E\to G$ is a homomorphism with respect to $\cdot$, ...
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Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map.

Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map. I've almost completed solving this problem, but ...
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Suppose $p:E\to B$ is a covering map and $B$ is connected. Prove that if $p^{-1}(\{b\})$ has n points $p^{-1}(\{b\})$ has n points for every $b\in B$

My idea is to somehow show that the group $O_n$ is both open closed which will imply $O_n=B$. Then assign to each $n$ the set of points $O_n\subseteq B$ such that $p^{-1}(b)$ has exactly $n$ points. ...
3
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1answer
52 views

Universal cover of boundary

Let $M$ be a compact manifold-with-boundary and $B$ a component of $\partial M$. Let $\tilde{M}$ be the univeral cover of $M$ with infinite-sheeted covering map $p:\tilde{M} \to M$. I wonder about the ...
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How does the fundamental group of the base space act on its universal cover?

I have a guess: Given $p : \tilde{X} \rightarrow X$, and fixing $x_0 \in X$, then $\pi_1(X, x_0)$ acts on $p^{-1}(x_0)$ in an obvious way. (Monodromy) Is this action the action that gives $X$ as a ...
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2answers
62 views

Finding open covers that do not contain finite subcovers

I'm being asked that for each of the following spaces $(X_i, T_i)$, find an open cover $U_i$ that does not contain a finite subcover. $X_i$ is a set and $T_i$ is a collections of subsets. I have ...
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1answer
68 views

Constructing covering space of surfaces

If $S_g$ is the surface $\#_g T^2$ where $g$ is a non-negative integer, when can we construct a covering space $S_h$ of $S_g$? Each such surface is a $CW$-complex, and in a $n$-sheeted covering, each ...
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1answer
27 views

What are the morphisms in the category of unramified coverings over a compact Riemann surface?

Fix a compact Riemann surface $S$, and finite a set of branch points $B \subseteq S$. Consider the collection of Riemann surfaces $S_1$ and mermorphic functions $f: S_1 \rightarrow S$, such that $f$ ...
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Algorithms for finding covering spaces of a given space

Taking the example of $X=S^1\vee S^1$ , to find the covering space $X$ what was done in Munkres is that we had the idea of how the real line wraps around the circle. Using this we attached circles ...
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1answer
75 views

Covering space of a $\theta$ graph

I'm considering a problem of finding an explicit algorithm to construct a covering of a finite graph (in particular, of a $\theta$ graph) Since the current graph is homotopy equivalent to a wedge of ...
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1answer
81 views

Geodesic Lines on Covering Maps

So I'm not sure how deck transformations work into this problem. I've established the following so far. Let $\pi:\tilde{M}\rightarrow M$ be the universal covering map. We may suppose that $M$ is ...
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1answer
112 views

Is there is a way to construct a covering space of a wedge of two circles for a given normal subgroup

I would like to construct a covering space of a wedge of two circles with a given normal subgroup $H \subset \pi_{1}(S^{1} \vee S^{1})=F_{a,b}$. The goal is to find a covering space $\tilde{X}$ so ...
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1answer
41 views

If p is a covering map of a connected space, does p evenly cover the whole space?

Suppose I have a covering map $p:X\rightarrow Y$, and $Y$ is connected. Is $Y$, as an open set, evenly covered by $p$? I think the answer is yes; I'm new to this kind of topology, so I'm not sure if ...
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1answer
68 views

Is There a Generalization of the Path Lifting Property Of Covering Maps.

$\newcommand{\R}{\mathbf R}$ Let $p:(E, e)\to (X, x)$ be a covering projection map. We know that for any path $\gamma:I\to X$ such that $\gamma(0)=x$, there is a unique lift $\Gamma:I\to E$ such that ...
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26 views

What happens if the Vector field vanishes in this case?

"Given the contravariant vector field $V{\mu}(x)$, we cosider te differential equation $$\frac{dx^{\mu}}{d\lambda}=V^{\mu}(x).$$ The solution $x^{\mu}$ is a map from $\mathbb{R}\rightarrow M$ is ...
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1answer
48 views

Étale morphism has all its Deck transformation homotopic to identity

Is there an example that étale morphism (of degree $d,d<\infty$) $\pi: X\rightarrow Y$, s.t. all its Deck transformations homotopic to $Id_X$,except the trivial one, where $Y$ is general Enriques ...
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35 views

Involution and Covering space

Is there a connected topological space such that admits a free involution, trivial fundamental group and furthermore has the set of real number as it's covering space?
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Catalog of covering maps

Is there somewhere where I can find a list of covering maps including their base space and target space? Apart from standard examples found in notes and books I can't find much else.
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80 views

Covering spaces of $S^1 \vee S^1$: to what subgroups do these ones correspond?

The universal covering space for $S^1 \vee S^1$ is the Cayley graph, $X$, of the free group on two generators, $F\{a,b\}$. The subgroup $F\{b\}$ corresponds to the covering space ...
2
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1answer
256 views

Deck transformations of universal cover are isomorphic to the fundamental group - explicitly

I have read on several places that given a (say path connected) topological space $X$ and its universal covering $\tilde{X}\stackrel{p}\rightarrow X$, there is an isomorphism ...
5
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1answer
121 views

Universal covering and double cover functors

Cross-posted on MO Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know ...
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1answer
82 views

Why $f:\mathbb{C} \to \mathbb{C},~~~ z \mapsto z^3$ is not a covering map? [closed]

Can someone tell me why $f:\mathbb{C} \to \mathbb{C},~~~ z \mapsto z^3$ is not a covering map?
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66 views

Covering Space Question

I recently encountered the following: Let $p:(E, e_0) \to (B, b_0)$ be a covering map. Assume that $p_∗(\pi_1(E, e_0)) \subseteq \pi_1(B, b_0)$ is a normal subgroup. If $e_1\in p^{−1}(\{b_0\})$, then ...
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3answers
104 views

Definition of compactness unnecessarily verbose?

The definition of a compact set is given as a set, $X$, for which all open covers have a finite subcover. This seems unnecessarily verbose to me. Wouldn't it be sufficient to simply say that $X$ has ...
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1answer
40 views

Make a complex polynomial a covering map

Let $p:\mathbb{C}\to \mathbb{C}$ be a complex polynomial. Let $C:=\{p(z):p'(z)=0\}$ and $V:=\mathbb{C}\setminus C$. I want to show that $p:p^{-1}(V)\to V$ is a covering map. By inverse function ...
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1answer
82 views

Proving that the tangent vector of a simple closed curve rotates by $ 2 \pi$

I am trying to prove that if $\gamma(t)=(x(t),y(t))$ ,a function from the closed interval $[0,1]$ to $\mathbb{R^2}$ is a simple closed unit speed curve such that $\gamma '(0)=\gamma '(1)$. Then the ...
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1answer
370 views

A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: ...
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1answer
61 views

Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Let $P\to $ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?
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1answer
116 views

What is a counterexample that a composition of covering maps not a covering map?

Let $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ be covering maps. What would be an example that $q\circ p:X\rightarrow Z$ is not a covering map? I saw a counterexample here, but it was too complex. Is ...
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1answer
63 views

The restriction fo covering to a component is a covering map onto its image.

I am reading Lee's Introduction to Topological Manifolds. I got stuck on the problem 11-7 on pages 303. The below is the problem. Prove : If $q: E \rightarrow X$ is a covering map and $A \subseteq ...
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1answer
43 views

Covering polygons with circles of minimal radius

I have a closed polygon and I would like to fully cover it with a set of K circles of different radius such that the area covered by the circles but outside the polygon is minimal. This seems the ...
5
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1answer
105 views

Analogy between Galois groups and fundamental groups

I've heard that there is an analogy between algebraic field extensions and covers (in topology). In this analogy Galois extensions correspond to Galois covers and Galois groups correspond to ...
5
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1answer
148 views

Example of a surjective local homeomorphism that is not a covering? [duplicate]

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering? I don't think this ...
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1answer
149 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
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Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
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1answer
113 views

Calculate the homology group of $S^3/G$, an Harvard qualifying exam problem with “unclear” solution

Problem Suppose that $G$ is a finite group whose abelianization is trivial. Suppose also that $G$ acts freely on $S^3$. Compute the homology groups (with integer coeffcients) of the orbit space ...
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Definition of covering (deck) transformation for smooth manifolds: Are they diffeomorphisms?

In John Lee's book Riemannian Manifolds, a covering transformation (or deck transformation) of a smooth covering map $\pi:\tilde{M}\to M$ (of connected smooth manifolds) is defined to be a smooth map ...
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29 views

Vitali covering over integers

Let $B(n,r)$ be a ball in $\mathbb{Z}^k$, that is, $B(n,r) = B'(n,r) \cap \mathbb{Z}^k$, where $B'(n,r)$ is a ball in Euclidean $k$-space. Suppose we have a set $W \subset \bigcup_{j=1}^N B(n_j,r_j)$. ...
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1answer
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Whether the fiber of a holomorphic covering of the unit disk over a non-simply-connected domain is infinite or not

Consider a holomorphic covering $f:\mathbb{D}\rightarrow \Omega$. Then for any point $a$ in the domain $\Omega$, consider the fiber $f^{-1}(a)$. If $f$ is non-constant, I know that when $\Omega$ is a ...
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How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
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1answer
73 views

A lift of isometry to universal covering

Let $M$ be a compact Riemannian manifold, $\bar M \to M$ be its universal covering and $\phi \in Isom(M)$ be an isometry of $M$. Is it true that, if $\phi$ is isotopic to the identity map of $M$, than ...
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1answer
110 views

Does there exist a double cover with trivial deck transformation group?

Sorry for the naive question. The following statement at the beginning of Bredon, chapter 4, §20, got me confused: Let $\pi:X \to Y$ be a two-sheeted covering map. Let $g:X \to X$ be the unique ...
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27 views

Name for measure of non-injectivity of a covering map

Suppose that $p:C\to X$ is a covering map. For $x\in X$, is there a name for the number $Card(p^{-1}(x))$? So that for $p(z)=z^5:\mathbb{C}\setminus\{0\}\to\mathbb{C}\setminus\{0\}$, one might say ...
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1answer
46 views

Restriction of complex polynomial that is a covering map

In my book the following exercise is given: let $p(z)\in \mathbb{C}[z]$ be a complex polynomial with distinct roots and degree $n>1$. Determine the greatest neighborhood $V$ of 0 such that ...
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1answer
64 views

Branched coverings of the Riemann sphere

Can someone give me an example of a non-trivial branched covering of the Riemann sphere? Is there some way to enumerate all such coverings? Is there any easy answer to the same questions about the ...
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Classification of Galois covering maps over a bouquet of 2 circles

The b-sheeted Galois covering maps over $C^*$ are equivalent to $z\mapsto z^b$. I wonder if there is an analogous statement for such Galois covers over C except two points $0,1$. Is that true that ...
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1answer
53 views

Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps). Let $f:M^m\to N^m$ be a smooth finite covering map. -- The following implication is not true: $M$ can be embedded ...
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84 views

If $G$ is a finite nontrivial group, then $K(G, 1)$ cannot be a finite CW-complex?

I am currently working on this algebraic topology problem and got stuck: Suppose $X$ is a finite CW complex with $\pi_1(X)$ a nontrivial finite group. Show that its universal cover $\widetilde{X}$ ...