For questions about or involving covering spaces in algebraic topology.

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4
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1answer
93 views

How can I prove that the hawaiian earring has no universal cover?

I know that the Hawaiian earring is not semi-locally simply connected so the existence is not guaranteed. Also, the point in which it must fail is the origin, where it isn't even locally simply ...
3
votes
1answer
87 views

Covering of hawaiian earring

I'm taking a course on Algebraic Topology and I'm struggling to find the solution to this problem: Let $Y$ be the Hawaiian earring in $\mathbb{R}^2$ and $Y'$ the union on infinite $Y$s moved $3z$ ...
0
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1answer
16 views

Example of non locally connected space with a covering in each connected component which is not a covering of the whole space

Can someone show me an example of an space $X$ non locally connected and another space $X'$ such that $\varpi: X' \to X$ is not a covering but $\varpi: \varpi^{-1}(X_i) \to X_i$ is, for each connected ...
2
votes
0answers
69 views

3 sheeted cover of Klein bottle with torus

So I'm dealing with this exercise in which it is asked to determine whether the torus can be a 3-cover of the Klein bottle. A friend of mine came up with a proof that this is not the case, but this ...
0
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1answer
24 views

Degree one branched cover is a homeomorphism

Suppose that $f:X \to Y$ is a branched cover of Riemann surfaces and a covering map of degree one outside of the ramification points. Then is $f$ a homeomorphism?
0
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1answer
30 views

The projection onto the orbit space $X/G$

Let $X$ be a locally compact, Hausdorff, path connected and locally path connected space. Assume a group $G$ acts freely and properly discontinuously on $X$, which means $\forall K^{compact},~~\{g\in ...
0
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1answer
29 views

Covering group $Aut(\tilde{X},p)\cong NH/H $

I want to show $Aut(\tilde{X},p)\cong NH/H $ where $H=p_*\pi_1(\tilde{X},\tilde{x_0})$ and $NH$ is the normalizer of $H$. In my text book, the author sketches the proof of the above by using the ...
1
vote
1answer
52 views

Definition of finite-sheeted covering

What is the definition of a finite-sheeted covering $q: E \to X$? Does it mean that every open $V \subseteq X$ has a pre-image $q^{-1}(V)$ that is the disjoint union of a finite number of sheets? Or ...
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0answers
29 views

Show entropy bound unit simplex

Let $\mathcal{S}_d$ be the $d$-dimensional unit simplex. Then for the norm $||x||_1 = \sum_i |x_i|$ and $0 < \varepsilon \leq 1$, $$N(\varepsilon, \mathcal{S}_d, ||\cdot||_1) \leq \left(\frac{5}{\...
1
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0answers
49 views

Find a nonregular 3-fold covering space of the genus two closed orientable surface.

Find a nonregular 3-fold covering space of the genus two closed orientable surface. This question was asked to me in Ph.D. Preliminary Exam. I have not any idea.
2
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2answers
44 views

How can I show that the composition of two coverings is also a covering?

I'm trying to prove the following: Let $\varpi ' : X'' \to X'$ and $\varpi : X' \to X$ be two coverings and let $X$ be a locally simply connected space. Prove that $\varpi \circ \varpi ' : X'' \...
2
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1answer
60 views

On a subgroup of the deck transformation of a covering space

I'm stuck with an exercise. Suppose you have a covering space $M \rightarrow X$, and you define $G:=\{\tau \in Deck(M)|\tau(S)=S\}$, for some $2$-sphere $S$ in $M$, and $G$ acts freely by isometries ...
0
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0answers
27 views

Show covering number $N(\epsilon,\mathcal{P},h) < \infty$ for all $\epsilon >0$

Let $\mathcal{P} = \{P_{\theta}: \theta \in \Theta\}$ be a dominated model of distributions on $[0,1]$. For the parameter space $\Theta$ we have $$\Theta := \{\theta: [0,1] \rightarrow \mathbb{R} \...
1
vote
1answer
39 views

Determine a normal covering space

Is there a way to determine if a covering space is normal without using the two theorems of Hatcher's book in pages 71 and 72?
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0answers
35 views

Covering map of the annulus

How to find universal covering map of the annulus of inner radius $\frac{1}{R}$ and outer radius $R>1$ from the right half plane $H$ where $H=\{z|Re(z)>0\}$?
0
votes
2answers
49 views

Size of the deck transformation group

If $p\colon Y\to X$ is a $k$-fold covering map, and $Y$ is path-connected, what is the size of Deck($p$), the deck transformation group? I was attempting to prove that the answer is $\leq k$, but ...
3
votes
1answer
63 views

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 2-...
2
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1answer
39 views

Covering a rectangle with circles

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2
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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 $\cdot$, ...
1
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0answers
46 views

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 ...
0
votes
2answers
55 views

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
votes
1answer
56 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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0answers
63 views

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 ...
2
votes
2answers
64 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 ...
1
vote
1answer
75 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 ...
0
votes
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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0answers
17 views

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 ...
2
votes
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 ...
2
votes
1answer
86 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 ...
1
vote
1answer
115 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 ...
3
votes
1answer
43 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 ...
2
votes
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 $...
1
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0answers
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 ...
1
vote
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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0answers
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?
0
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0answers
14 views

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.
0
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0answers
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
votes
1answer
310 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 $$\mathrm{Deck}(\tilde{...
5
votes
1answer
122 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 ...
1
vote
1answer
84 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?
0
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0answers
67 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 ...
0
votes
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 ...
1
vote
1answer
41 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 ...
6
votes
1answer
83 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 ...
4
votes
1answer
422 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: If ...
1
vote
1answer
67 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?
0
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1answer
121 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 ...
3
votes
1answer
66 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 ...
3
votes
1answer
46 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
votes
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 ...