For questions about or involving covering spaces.

learn more… | top users | synonyms

3
votes
1answer
76 views

Is $f$ necessarily a covering?

Let $f : X \rightarrow Y$ be a continuous map of spaces where $X$ is compact Hausdorff , $Y$ is Hausdorff and both spaces are path-connected and locally path-connected. Suppose that for every $x \in ...
4
votes
1answer
83 views

Restriction of Covering Space

I'm studying for an exam, and got stuck on the following exercise: Find all two-sheeted covering spaces for $X =\mathbb{S}^1 \vee \mathbb{S}^1$. Label the two circles of $X$ by $a$ and $b$. Attach ...
5
votes
1answer
249 views

The action of the group of deck transformation on the higher homotopy groups

This is for homework. I'm supposed to do exercise 4.1.4 in Hatchers "Algebraic Topology", which is to show that given a universal covering $p: \tilde{X} \to X$ of a path-connected space $X$, the ...
0
votes
0answers
55 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
2
votes
1answer
90 views

All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
7
votes
2answers
341 views

A locally constant sheaf on a locally connected space is a covering space; Proof?

As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
1
vote
1answer
60 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
0
votes
1answer
51 views

Plane rotation homomorphism

I want to show that the following set (given group structure with matrix multiplication) $$D=\left\{ D(\theta)=\begin{pmatrix} e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2} \end{pmatrix}, 0 \leq ...
2
votes
3answers
128 views

Relationship between the fundamental group and the natural equivalence classes of its universal cover

For a universal covering $p: Y \to X$, under the equivalence relation $y_1 \sim y_2$ if $p(y_1) = p(y_2)$, $Y$ admits the quotient map $\, \, \, q: Y \to Y / \sim$. There is a natural bijection $\bar ...
2
votes
1answer
56 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
4
votes
2answers
118 views

Extending a quotient map to a covering map on $\mathbb{RP}^2$

Why can we not extend the quotient map $q:[0,1]\times[0,1] \to \mathbb{RP}^2$ to a covering map, $\mathbb{R}^2 \to \mathbb{RP}^2$?
2
votes
2answers
252 views

Universal cover of a figure eight?

An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'. Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can ...
2
votes
1answer
107 views

Universal cover modulo the monodromy action

Let $\tilde{X} \to X$ be the universal cover of a connected, locally path-connected and semi-locally simply connected topological space $X$. Is it always true that the orbit space $$ \tilde{X} \;/\; ...
2
votes
2answers
129 views

Is the universal covering surface orientable?

Let $M$ be a smooth, say also closed (compact and without boundary) surface. Is it true that its universal covering surface is orientable?
3
votes
1answer
126 views

Domain is Hausdorff if image of covering map is Hausdorff

Suppose that $p:X\rightarrow Y$ is a covering map. Show that if $Y$ is Hausdorff, then so is $X$. I have an answer but I'm not sure if it's right? By definition of Hausdorff, $\forall x,y, \in Y, ...
2
votes
0answers
156 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then ...
2
votes
1answer
260 views

Domain is locally path-connected if image of a covering map is locally path-connected.

Show that if $p:X\rightarrow Y$ is a covering map and $Y$ is locally path-connected, then so is $X$. How do you go about proving this? I can think of two ways of doing this, either by definition of ...
3
votes
2answers
196 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
0
votes
1answer
39 views

Defining a homeomorphism from an equivalence relation on a covering map domain to its image.

If $p: \tilde Y \to Y$ is a covering map, and we define an equivalence relation $\sim$ on $\tilde Y$ by $\tilde y_1 \sim \tilde y_2$ if $P(\tilde y_1) = p(\tilde y_2)$. How would you show that the ...
0
votes
1answer
84 views

Showing the fibre over a point in a covering map is a discrete space.

If $ p : \tilde Y \to Y$ is a covering map, how would you show that for every $y \in Y$ we have that $p^{-1}(y)$ is a discrete space?
2
votes
0answers
59 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
1
vote
0answers
75 views

Question on homotopy lifting

I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear. A lemma in my lecture notes says: Let $p: \tilde Y \to ...
3
votes
2answers
198 views

Why must a finite covering map be closed?

A covering map $p:C\to X$ is finite when for each $x\in X$ we have $|p^{-1}(x)|<\infty.$ I have to prove that such a covering map has to be closed. I'm having trouble with it. When $p$ is a ...
2
votes
0answers
102 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
5
votes
1answer
99 views

Classifying a Branched Covering Space

This question comes from the proof of proposition 2.2 in Henry Laufer's 'Normal Two-Dimensional Singularities" text. I am excerpting the part I don't understand, and I think it's a self-contained ...
3
votes
2answers
87 views

Is this covering map a homeomorphism?

Suppose $\pi : \widetilde X \to X$ is a finite, connected covering, and suppose that there exists a continuous map $f: \widetilde X \to \mathbf R^2$ which is injective on each fibre of $\pi$. Is $\pi$ ...
0
votes
2answers
144 views

How to prove that $p:S^1\rightarrow S^1$ $z\mapsto z^2$ is a covering map?

How can I prove that $p:S^1\rightarrow S^1$, $z\mapsto z^2$ is a covering map? Please help. I was not able to prove it by applying definition of covering space.
14
votes
1answer
371 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
4
votes
3answers
277 views

proving that a covering map with certain domain and range is homeomorphism

Let $p:E\to B$ be a covring map, with $E$ path connected. Show that if B is simply connected, then $p$ is a homeomorphism. Well I don't know exactly what can I do here, maybe I have to start with ...
4
votes
1answer
309 views

Covering space homeomorphism

In the course of an exercise from Hatcher's topology text, I came to the following point. Given $p: \tilde{X} \to X$ the universal cover for $X$, and a continuous map $h: \tilde{X} \to \tilde{X}$ ...
3
votes
1answer
385 views

covering map with finite fibres and preimage of a compact set

Let $f:X\to Y$ be a covering map (covering maps are surjective) , Y be compact set. And suppose that $f^{-1}(y) $ is finite for each $y\in Y$. Prove that $X$ is also compact. I think that this ...
2
votes
0answers
94 views

Characterization of maps with $\mathbb{Z}/2\mathbb{Z}$-equivariant lifts.

I'm interested in characterizing maps $f:\mathbb{R}P^k\to\mathbb{R}P^\infty$ that lift to a $\mathbb{Z}/2\mathbb{Z}$-equivariant map $\tilde{f}:S^k\to S^\infty$. For $k\geq 2$ I have been able to ...
6
votes
2answers
247 views

Orientable double covers for non-orientable manifolds

If I have two non-orientable connected manifolds such that their orientable double covers are homeomorphic, can anything be said about the manifolds? Are they homeomorphic?
4
votes
1answer
278 views

composition of certain covering maps

This problem was posted before, but not the proof (because the asker knowed the answer), only a counterexample without the hypothesis of finite fibres. I want to know how to prove this proposition: ...
1
vote
1answer
153 views

preimage of a connected under a covering map has unique representation into slices

Let $p:E\to B$ be a covering map. Suppose that $U$ is an open set of $B$ that is evenly covered by p. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ into slices is unique. I have no ...
10
votes
1answer
362 views

Prove that a covering map is a homeomorphism

I got stuck in the following exercise: Let $p:\widetilde{X}\rightarrow X$ be a covering map with $\widetilde{X}$ connected and $p^{-1}(x)$ finite, for every $x\in X$. Show that if there exists a ...
2
votes
1answer
46 views

Varieties with infinitely many topological covers of finite degree

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers. If $\dim X =1$, this holds if and only if the genus of $X$ is positive. Do we have a similar ...
4
votes
0answers
109 views

An entire function with finite covering group is a polynomial.

Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
0
votes
2answers
162 views

Lifting an automorphism to the universal covering space..

Let $X$ be a manifold and $Y$ be its universal covering. Is it true that any $\phi \in \mathrm{Aut}(X)$ can be lifted to $\overline{\phi}\in \mathrm{Aut}(Y)$?
1
vote
0answers
34 views

Classification theorem of the coverings of a given space

I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as Classification of covering spaces ...
1
vote
0answers
49 views

How to find every 4-sheet covering of the wedge sum?

Based on this question 4-sheet covering of the wedge sum of two circles I know how to find one 4-sheet covering of the wedge sum, but how to find every 4-sheet covering of the wedge sum? I really ...
0
votes
0answers
88 views

Covering space (Lie groups and their maximal tori)

Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
2
votes
0answers
31 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
2
votes
0answers
57 views

Global sections of covering spaces

Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$. Is there any reference where this is explicitly ...
4
votes
2answers
438 views

Covering space of a non-orientable surface

I have the following problem: Find the 2-sheeted (orientable) cover of the non-orientable surface of genus g. The cases $g=1,2$ are well-known, we have that the cover of $\mathbb{R}P^2$ is ...
1
vote
1answer
146 views

The definition of normal covering in Hatcher book

In page 70 of Hatcher's book, in the section Deck Transformations and Group Actions, the author defines a normal covering in the following way: A covering space $p:\tilde X\to X$ is called normal ...
1
vote
1answer
111 views

How can I find the deck transformations of $p:\mathbb R\to S^1$?

How can I find the deck transformations of $p:\mathbb R\to S^1$, where $p(t)=e^{2\pi ti}$? I tried in this way: Let $\phi:\mathbb R\to \mathbb R$ be a deck transformation, so $p(\phi(t))=p(t)$ for ...
4
votes
1answer
119 views

What's the automorphism group of this covering?

What's the automorphism group of this covering? I know why this is a covering, but I don't know how to find the automorphism group of this covering. I need help, thanks
2
votes
1answer
68 views

Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
0
votes
3answers
59 views

Is every fiber-preserving map between coverings again a covering?

Suppose we have two coverings $$p_1:Y\rightarrow X$$ and $$p_2:Y^\prime\rightarrow X$$ and further a continuous map $$\pi\colon Y\rightarrow Y^\prime,$$ such that $$p_2\circ\pi=p_1.$$ Ist $\pi$ ...