The covering-spaces tag has no wiki summary.
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Covering space Hausdorff implies base space Hausdorff
There is an exercise problem in Hatcher's Algebraic Topology book asking to show that if $p:\tilde{X}\rightarrow X$ is a covering space with $p^{-1}(x)$ finite and nonempty for all $x\in X$, then ...
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The covering space of a region contained in complex plane delete two points.
We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
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1answer
30 views
The restriction of a covering map on the connected component of its definition domain
Suppose $p:Y\to X$ is a covering map, $X,Y$ are manifolds and $X$ is connected. If $Z$ is a connected component of $Y$, I wonder if the restriction of $p$ on $Z$ is also a covering map? If not, what ...
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Riemann surface arising as a quotient of the upper half-plane.
Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$
Denote $\Gamma$ the ...
5
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1answer
73 views
Induced map on homology from a covering space isomorphism
Suppose $S^1 \times \mathbb{R}P^2$ covers some space. Why is it that any covering space isomorphism $h$ induces the identity map on $H_1$? I don't see how to prove this except maybe from looking at ...
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1answer
58 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 ...
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1answer
36 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 ...
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35 views
Which of the following spaces nontrivially cover themselves?
I am having some difficulties with a qualifying exam question. I would appreciate if someone could give me a little help.
Which of the following spaces nontrivially cover themselves?
(a) $S^3$
(B) ...
5
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1answer
79 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 ...
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35 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.
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1answer
45 views
Covering infinite sheeted covering of torus
Suppose I have subgroup $H=\operatorname{span}\langle (a,b)\rangle\subset \pi_1(\mathbb{T}^2)=\mathbb{Z}^2$, where $a,b$ are integers where $(a,b)\neq(0,0)$. I know the covering space is $S^1\times ...
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1answer
39 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 ...
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20 views
algebraic structure of group of covering morphisms between covering spaces?
I need an answer to the following problem, but have been told to post on here instead of thinking about it!
Let $X$ satisfy the usual assumptions, $p_1:Y_1\rightarrow X$, $p_2:Y_2\rightarrow X$ ...
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60 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 ...
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40 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 ...
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1answer
28 views
Plane rotation homomorphism
I want to show that the following set (given group structure with matrix multiplication)
$$D=\left\{ D(\theta)=\begin{pmatrix}
e^{i\theta/2} & 0 \\
0 & e^{-i\theta/2}
\end{pmatrix}, 0 \leq ...
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Relationship between the fundamental group and the natural equivalence classes of its universal cover
For a universal covering $p: Y \to X$, under the equivalence relation $y_1 \sim y_2$ if $p(y_1) = p(y_2)$, $Y$ admits the quotient map $\, \, \, q: Y \to Y / \sim$. There is a natural bijection $\bar ...
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Number of sheets of Covering space of $\mathbb{T}^2$ after transformation by $SL(2,\mathbb{Z})$
Suppose you have a subgroup of $H\subset\pi_1(\mathbb{T}^2)=\mathbb{Z}\times\mathbb{Z}$, $H=\text{span}\langle u,v\rangle$. If you have an element $G\in SL(2,\mathbb{Z})$, do the covers of ...
2
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1answer
35 views
A restricted continuous map is a homeomorphism
Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
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Extending a quotient map to a covering map on $\mathbb{RP}^2$
Why can we not extend the quotient map $q:[0,1]\times[0,1] \to \mathbb{RP}^2$ to a covering map, $\mathbb{R}^2 \to \mathbb{RP}^2$?
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2answers
70 views
Universal cover of a figure eight?
An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'.
Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can ...
2
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1answer
39 views
Universal cover modulo the monodromy action
Let $\tilde{X} \to X$ be the universal cover of a connected, locally path-connected and semi-locally simply connected topological space $X$. Is it always true that the orbit space
$$ \tilde{X} \;/\; ...
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1answer
35 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?
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1answer
65 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, ...
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74 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 ...
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1answer
108 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 ...
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2answers
110 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 ...
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1answer
32 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 ...
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1answer
33 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?
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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 ...
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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 ...
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2answers
70 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 ...
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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. ...
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1answer
63 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 ...
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61 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 $\pi$ ...
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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.
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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 ...
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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
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1answer
108 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 $h: \tilde{X} \to \tilde{X}$ ...
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1answer
75 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 ...
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26 views
Pullback of a family of curves via a covering map.
Let $X$ be a smooth compact projective manifold and $\pi:Y\rightarrow X$ a Galois covering map. Is it always possible to pull back a family of curves on $X$ to $Y$?
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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 ...
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148 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?
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1answer
79 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:
...
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1answer
52 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 ...
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1answer
141 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 ...
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1answer
34 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 ...
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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 ...
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1answer
52 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 $\overline{\phi}\in \mathrm{Aut}(Y)$?
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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 ...
