For questions about or involving covering spaces in algebraic topology.

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7
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2answers
146 views

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 $...
4
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1answer
60 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 ...
2
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0answers
47 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
votes
1answer
79 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 ...
5
votes
2answers
126 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 ...
0
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0answers
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 "...
0
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1answer
47 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 $p:p^{-1}...
0
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1answer
65 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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0answers
44 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 ...
8
votes
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 ...
3
votes
2answers
87 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
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2answers
89 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 ...
3
votes
1answer
71 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
votes
2answers
32 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 ...
4
votes
3answers
190 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 ...
1
vote
1answer
81 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
36 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 $\mathfrak{i}(M,\mathrm{g})$...
2
votes
0answers
199 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 well ...
1
vote
1answer
60 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
44 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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1answer
40 views

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

Let $G=\mathbb Z\rtimes\mathbb Z$ and $H=\mathbb Z\rtimes7\mathbb Z$. 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 \mid (m,n)(l,7k)(m,n)^{-...
8
votes
2answers
140 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 $\operatorname{...
3
votes
1answer
62 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
82 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
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1answer
66 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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0answers
29 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?
0
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2answers
90 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 ...
1
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1answer
82 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).
2
votes
2answers
77 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 ...
3
votes
3answers
66 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} : \...
4
votes
2answers
125 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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0answers
31 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 ...
1
vote
1answer
36 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}, f|_{V_2}...
4
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1answer
257 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. ...
3
votes
1answer
93 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
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1answer
47 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
94 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 ...
2
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1answer
172 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
2
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3answers
66 views

locally path connectedness

While studying covering spaces , hatcher mentioned the "shrinking wedge of circles" this space is locally path connected as I was told , but I wasn't able to prove it nor to see it, it looks like comb ...
2
votes
1answer
100 views

Universal covering of the complement of a circle in $\mathbb{R}^3$

What is the universal covering of $X=\mathbb{R}^3\setminus(S^1\times\{0\})$? I've been trying to build a covering map from $\mathbb{R}^3$ onto $X$ via composition of $p:\mathbb{R}^3\to Y$ and $q:Y\to ...
1
vote
1answer
246 views

Algorithm - Circle Overlapping

Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape ...
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vote
2answers
371 views

Finite fundamental group and covering spaces

Show that if a path-connected, locally path-connected space X has a finite fundamental group , then every map $X$ to $S^1 \times S^1$ is nullhomotopic (i.e. homotopic to a constant map) . Is the ...
2
votes
1answer
184 views

Constructing a simply connected covering space

"Construct a simply connected covering space of the space that is the union of the sphere S2 with two of its intersecting diameters." can anyone help me with this? i don't know how to think , all ...
0
votes
2answers
99 views

Explain why the following statement is false

Let $f:S^1 \to S^1$ be given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$. Then there is a unique lift $\bar f: S^1 \to \mathbb{R}$ with the properties that (i) $\bar f(1)=0$ and (ii)...
1
vote
1answer
65 views

12.16 in Lee's Introduction To Topological Manifolds

Reading through Lee's Introduction To Topological Manifolds. Theorem 12.16 says the following: Suppose G and H are connected, locally path-connected topological groups, and $\phi:G \to H$ is a ...
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1answer
52 views

Is the tangent bundle of a covered manifold a quotient manifold?

Given a covering manifold $\rho :\widetilde M \to M$ we know that $M$ can be thought of as the quotient space of $\widetilde M$ like so $M = \widetilde M /\ G$ where $G$ is the monodromy group (or ...
2
votes
4answers
249 views

Longest chord in the intersection n disks (circle areas)

Given n disks that intersect, there is a shape in the space where they intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
0
votes
2answers
144 views

Is the Riemann surface for the square root simply connected?

I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...
2
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0answers
42 views

About prime geodesic cycles and deck transformations group

I'm proving theorem 2 occurring in Sunada's paper Riemannian coverings and isospectral manifolds. Unfortunately Sunada's quotes himself to the following paper: Tchbotarev’s density theorem for closed ...