For questions about or involving covering spaces.

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0
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1answer
62 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
0
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1answer
40 views

For a finite etale map $X\rightarrow T$ of degree $d$, and a $U$-point $t\in T(U)$, are there at most $d$ points in $X(U)$ lying over $t$?

If $f : X\rightarrow T$ is a finite etale morphism of connected schemes, and $U$ is another connected scheme and we're given a map $t : U\rightarrow T$, then must it be true that there are at most $d$ ...
6
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1answer
117 views

Deck transformations on $S^1\times \mathbb{R}P^2$

I'm studying for qualifying exams and stuck on the following problem: Suppose that $S^1\times \mathbb{R}P^2$ covers a space, and let $h$ be a deck transformation of the covering. Show that the ...
2
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0answers
77 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if ...
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0answers
57 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
0
votes
1answer
89 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
4
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1answer
69 views

Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
7
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1answer
146 views

Compact subspace of a covering space

I've been working through Massey's A Basic Course in Algebraic Topology and I've gotten stuck on the following exercise (V.8.4): Let $X$ be a regular topological space, and $(\tilde{X}, p)$ a ...
0
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2answers
47 views

Liftings in Topology

I am wondering about this: Assume you have a class $[f] \in \pi_1(B,b_0)$ and a covering map $p:E \rightarrow B$. Now, I know that if you take any two paths $g,h \in f$ that are homotopic and they ...
4
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2answers
89 views

Group of automorphisms of a compact hyperbolic Riemann surface is finite

Let $M$ be a compact hyperbolic Riemann surface. Is there a simple way to show that the automorphism group $Aut(M)$ of conformal self-mappings of $M$ is a finite group? Recall that a hyperbolic ...
2
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0answers
44 views

Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
4
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1answer
61 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
2
votes
1answer
22 views

How morphism of coverings induces morphism of automorphisms groups?

Let $X_1\to Y$ and $X_2\to Y$ be two coverings. How does a morphism $X_1\to X_2$ over $Y$ induce morphism $\operatorname{Aut}_Y(X_1)\to \operatorname{Aut}_Y(X_2)$? It should be trivial, but I can not ...
1
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1answer
68 views

Action of $\mathbb Z_2$

Is there a connection between Artin-Schreier theorem on finite groups which can be absolute Galois groups and the classification of finite groups freely acting on even-dimensional sphere? The former ...
1
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1answer
48 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
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0answers
56 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 ...
1
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1answer
29 views

Size of minimal covering, overlapping and disjoint

There are two ways to cover a geometric shape with primitive units: you can allow the units to overlap, or require that they be disjoint. Of course the number of units in the case of disjoint covering ...
0
votes
1answer
69 views

What are the deck transformations of this threefold cover of the figure 8?

Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators $a$ and $b$ and the relations $a^2, b^2, aba^-1, bab^-1$. I thought ...
1
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0answers
177 views

Atiyah & Macdonald's Introduction to Commutative Algebra, Exercise 8.5

The exercise asks the reader to prove that $X$ is a finite covering (i.e., the number of points of $X$ lying over a given point of $L$ is finite and bounded) of $L$, where the affine varieties $X$ and ...
3
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2answers
219 views

A proper local diffeomorphism between manifolds is a covering map.

The following is an exercise taken from "Manifolds and Differenial Geometry" by Jeffrey M. Lee. Let $\widetilde M$ and M be (connected) $C^r$ manifolds. Let $f: \widetilde M \to M$ be a proper map ...
3
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1answer
45 views

The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
4
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1answer
82 views

Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
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0answers
56 views

Show That a Lift Always Exists

I've been considering this problem: Suppose that $X$ is a topological space and that $H_1(X)$ is a finite group of odd order. Show that if $p:\tilde{Y}\rightarrow Y$ is a covering space of index ...
0
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1answer
65 views

Are deck transformations homotopic to the identity?

Suppose that $p: X \to Y$ is the universal covering of some connected and locally path connected space $Y$, and that $\phi$ is a deck transformation. Is $\phi$ homotopic to the identity on $X$? If so, ...
2
votes
1answer
94 views

Generalisation of Vitali's covering lemma

In "The geometry of fractal sets", Falconer gives the following generalisation of the Vitali covering lemma as an exercise: Let $\mu$ be any measure on $\mathbb{R}^{n}$ and $E$ a set with ...
0
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0answers
80 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
3
votes
1answer
99 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
3
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3answers
111 views

covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
0
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1answer
55 views

Lifting property of a covering space

A common example of a covering space is $p:\mathbb R \rightarrow S^1 $, $p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ...
2
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0answers
45 views

lifting injective maps to injective maps in principal bundles

Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde ...
1
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1answer
39 views

Branched covering of a manifold [duplicate]

What would be the definition of a branched covering of a manifold? I am not familiar with branched coverings at all.
2
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1answer
65 views

References for a standard result about coverings of Riemann surfaces

I my thesis I have to cite the following standard result: Let $Y$ be a compact Riemann surface and let $B\subseteq Y$ be a finite subset. Given a natural number $d$, there are only finitely many ...
3
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1answer
32 views

Help proving a space is closed in order to show a space is properly discontinuous

This stems from exercise 6, section 81 in Munkres. Let $X$ be a locally compact Hausdorff space; let $G$ be a group of homeomorphisms of $X$ such that the action of $G$ is fixed-point free. Suppose ...
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0answers
26 views

Composition of covering maps is a covering map if the inverse image is finite. [duplicate]

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 $\forall z$, then $p$ is a covering map. $\textbf{My Attempt:}$ Let $U$ be an arbitrary open ...
0
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1answer
46 views

Why is this induced homomorphism not surjective?

Let $p: Y \to X$ be a covering with fix base point. I have already shown that the induced homomorphism $p_{*}=\pi_1(Y,y_0) \to \pi_1(X,x_0)$ is injective. However, since we are not calling it an ...
0
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0answers
44 views

lifting a closed curve

Is it always true (because of covering spaces has homotopy lifting property)? loop $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve. or we have to ...
3
votes
1answer
75 views

Showing that every map $f : S^2 \rightarrow S^1$ is homotopic to the trivial map

Problem: Prove that every map $f : S^2 \rightarrow S^1$ is homotopic to the trivial map. Hint: Use the covering space $E: \mathbb{R} \rightarrow S^1$. If you can show that every map $f: ...
0
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0answers
60 views

Lifting triangulations to universal covers

I thought I would have been able to find more information about this by simply googling than I have been; suppose I have the information that $X$ is constructed by taking a finite disjoint collection ...
1
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1answer
59 views

Why is $p$ regular?

This is with regards to this lemma. Lemma: Let $p: \tilde{X} \rightarrow X$ be a covering map. Assume $\tilde{X}$ is connected and locally path-connected. Then $p$ is a regular covering $\iff$ for ...
1
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1answer
58 views

let $p: E\rightarrow B$ conitnuous and surjective. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices is unique.

let $p: E\rightarrow B$ coninuous and surjective and suppose that $U$ is an open set of $B$ that is evenly covered by p and U. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices ...
6
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0answers
131 views

Are there infinitely many non-negative integers not covered by one of these 7 polynomials?

Consider the following polynomials: $$ \begin{align} f_1(n, m) &= 30nm + 23n + 7m + 5 \\ f_2(n, m) &= 30nm + 17n + 13m + 7 \\ f_3(n, m) &= 30nm + 23n + 11m + 8 \\ f_4(n, m) &= ...
3
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2answers
86 views

How can i find this universal cover? [closed]

I have $X = \{(x,y,z) | x^2 + y^2 = 1, z = 0\}$ and $Y = \{(x,y,z)| (x-1)^2 + y^2 + z^2 = 1\}$. What is the universal cover of $X \cup Y$?
3
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0answers
80 views

Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
1
vote
2answers
114 views

lifting a product of commutators of standard generators on 2-manifolds

I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf I don't understand this part: "(...) we can easily ...
3
votes
0answers
44 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
1
vote
1answer
45 views

Let $p$ be a covering space and $X, Y$ be path connected. Show there exists a map $q$ such that $q\circ p=1_{X}$ iff $p$ is a homeomorphism.

Let $p\colon X\rightarrow Y$ be a covering map where $X$ and $Y$ are path connected. Show that there exists a map $q\colon Y\rightarrow X$, such that $q\circ p=1_{X}$ if and only if $p$ is a ...
3
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0answers
66 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
4
votes
1answer
123 views

Show that if B is simply-connected, then p is a homeomorphism.

Let $p: E \rightarrow B$ be a covering map with $E$ path-connected. Show that if $B$ is simply-connected, then $p$ is a homeomorphism. I'm checking to see if my solution is flawed. Since $p$ is a ...
1
vote
2answers
193 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
12
votes
4answers
403 views

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...