For questions about or involving covering spaces in algebraic topology.

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5
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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 ...
2
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1answer
1k views

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

Given a rectangle of area $n\times m$ and identical circles with radius $r$. What is the optimal solution for covering this rectangle with minimum number of circles? I found a relative solution here. ...
5
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3answers
687 views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
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0answers
24 views

A manifold is a covering space over its quotient by a group action

Let $M \times G\to M$ be a properly discontinuous, free action of group $G$ on a manifold $M$. The quotient topology of the orbit space is Hausdorff. Suppose $p\in M$. How can we choose an open ...
2
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2answers
38 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $f^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
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0answers
19 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 $W \subset \bigcup_{j=1}^N B(n_j,r_j)$. ...
4
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1answer
40 views

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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0answers
18 views

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 ...
4
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1answer
22 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 ...
2
votes
1answer
139 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
4
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1answer
43 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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0answers
22 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 ...
0
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1answer
26 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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0answers
36 views

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 ...
0
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1answer
37 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 ...
8
votes
1answer
39 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 ...
2
votes
3answers
125 views

Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
4
votes
2answers
64 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 $\widetilde{X}$ ...
2
votes
2answers
62 views

A space $X$ with $S^{2n+1}$ as universal covering space must be orientable.

Problem If $X$ has $S^{2n+1}$ as universal covering space, then show that $X$ must be orientable. My idea: By contradiction, suppose $X$ is non-oreintable. Then we consider the orientation covering ...
2
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2answers
28 views

A space homeomorphic to the connected sum $\mathbb{RP}^3$ # $\mathbb{RP}^3$

Problem (1) Consider the space $Y$ obtained from $S^2 \times [0,1]$ by identifying $(x,0)$ with $(-x,0)$ and also identifying $(x,1)$ with $(-x,1)$, for all $x\in S^2$.Show that $Y$ is ...
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1answer
19 views

Existence of the universal covering space of a connected Lie group

I am working on a project about how the universal cover of a connected Lie group is a Lie group, but I cannot find a theorem that assures that this universal cover actually exists. I've found ...
2
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0answers
63 views

Show that if $f$ is a proper surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is ...
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1answer
77 views

Covering spaces of $S^1$

Put $\tilde X=\lbrace (exp(2\pi if(t)),t)| t\in \mathbb{R} \rbrace$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is any continuous function and let $\pi_1$ be the projecction on the first coordinate. ...
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0answers
29 views

Isometry algebra implication from Riemannian covering

I really wish that, if $\pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h})$ is a Riemannian covering, then $\mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g})$, where ...
1
vote
1answer
50 views

An exhaustive continuous map is a covering map.

$p_1:\tilde X_1 \rightarrow X \, ; \, p_2:\tilde X_2 \rightarrow X$ two coverings maps, where $X$ connected and locally path-connected, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
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0answers
20 views

Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
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0answers
34 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
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1answer
37 views

Need to check if $H\triangleleft G$ in a covering of the klein bottle

Let $G=Z\times Z$ and $H=Z\times7Z$. I want to check if H◁G. I know I need to calculate N_G(H), and I think this is $$N_G(H)={(m,n) ∈ G | (m,n)(l,7k)(m,n)^{-1} ∈ H \forall l,k}$$ but I'm not sure ...
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2answers
94 views

$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
3
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1answer
44 views

For a covering map, if the target space is Hausdorff, so is the source

I am working on proving that if $q:E\rightarrow X$ is a covering map, and $X$ is Hausdorff, then so is $E$. The answer to this question: Domain is Hausdorff if image of covering map is Hausdorff ...
2
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2answers
52 views

Find all covering spaces of $\mathbb{RP}^n \times \mathbb{RP}^n$, $n>1$

Let $X = \mathbb{RP}^n \times \mathbb{RP}^n$. I know the following: the universal cover of $X$ is $Y = \Bbb S^n \times \Bbb S^n$ the fundamental group of $X$ is $G = \Bbb Z/2 \Bbb Z \times \Bbb Z/2 ...
2
votes
3answers
200 views

Is a path connected covering space of a path connected space always surjective?

If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where ...
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0answers
74 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
2
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1answer
52 views

If $f\circ g$ is continuous and $f$ is a local homeomorphism, then $g$ is continuous

Suppose $g:X\to Y$ and $f:Y\to Z$, and $f$ is a local homeomorphism, which is to say that for any $y\in Y$ there is a neighborhood $U$ of $y$ such that $f\restriction U$ is a homeomorphism from $U$ to ...
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1answer
71 views

Covering $S_2$ with $S_3$(or $S_n$)

How can I construct a covering map $p : S_3 \to S_2$? I can construct coverings for $S^1\vee S^1$, but same ideas don't work for $S_2$ ($S_g$ is sphere with $g$ handles).
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0answers
24 views

Automorphisms of simple covers of Riemann surfaces

Can anybody give me a simple proof that simple covers of a Riemann surface have no covering automorphisms?
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2answers
61 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
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1answer
644 views

Klein bottle covered by the torus

Maybe this is an idiot question and I'm missing something very trivial. This question question was asked here before, but the answer (which apparently is equal to the one that I created) seems ...
2
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2answers
44 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
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2answers
111 views

Finding the Fundamental Groups of Some Modular Spaces

I'm looking to compute the fundamental group of a couple of different quotients of the $n$-torus. The first of these I'm interested is the space $\mathbb{T}^n/S_n$ where the symmetric group $S_n$ ...
2
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3answers
51 views

simply connected covering of a path connected space (II)

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
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1answer
202 views

Question on covering spaces

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 space. I'm confused that $p=r\circ q$ is obvious ...
3
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2answers
99 views

Is composition of covering maps covering map?

In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that ...
3
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2answers
115 views

A problem on covering space from Hatcher book…

I was trying a problem from Hatcher's book Algebraic Topology, in section 1.3 problem number 12. Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. ...
4
votes
2answers
420 views

If a covering map has a section, is it a $1$-fold cover?

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
28 views

Section of a covering projection from a connected space [duplicate]

Let $p:\overline{X}\rightarrow X$ is a continuous mapping. A continuous map $s:X\rightarrow \overline{X}$ such that $p\circ s =Id_X$ is called a section of $p$. Suppose $\overline{X}$ is connected ...
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1answer
31 views

if $V_1\cong U_1, V_2\cong U_2$, is $(V_1\cup V_2 \cong U_1\cup U_2)$? Pasting homeomorphisms

My question arises from the theory of covering spaces. assume $f:Y\to X$ is a covering map, or more generally a local homeomorphism. Assum $U_1,U_2\subset X$ are open sets such that $f|_{V_1}, ...
2
votes
1answer
77 views

simply connected covering of a path connected space

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
1
vote
1answer
33 views

About covering spaces

Suppose X is a topological space whose fundamental group is Z x Z x Z2 x Z3. Is it possible for the wedge sum of two circles to be a covering space for X? Can anyone help me with this ?
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1answer
47 views

Prove that a non-empty subset of an open set which is evenly covered is evenly covered

Let $p: E\rightarrow B$ a continuous surjective map and $U \subseteq B$ be open and not empty and who is being evenly covered by $p$. Show that all non-empty subsets of $U$ are being evenly covered by ...