For questions about or involving covering spaces.

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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 ...
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1answer
27 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 ...
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1answer
65 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 ...
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135 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 ...
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2answers
192 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 ...
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1answer
41 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 ...
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1answer
79 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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53 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 ...
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1answer
61 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, ...
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1answer
85 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 ...
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76 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 ...
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94 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 ...
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3answers
107 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 ...
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1answer
49 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 ...
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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 ...
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1answer
34 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.
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1answer
64 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 ...
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1answer
31 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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25 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 ...
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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 ...
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42 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 ...
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1answer
72 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: ...
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53 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 ...
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1answer
58 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 ...
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1answer
51 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 ...
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130 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) &= ...
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2answers
82 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 $X \cup Y$?
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74 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 ...
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2answers
113 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 ...
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40 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 ...
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1answer
44 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 ...
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63 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 ...
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1answer
116 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 ...
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2answers
182 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 ...
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377 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 ...
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1answer
34 views

Verify $p _0 : [0,1] \mapsto S^1 , p_0(s)=(\cos(2 \pi s),\sin(2\pi s))$ is a covering map.

I want to verify that the restriction to the interval $[0,1]$ of the map $p : \mathbb{R} \mapsto S^1 $ given by $ p(s)=(\cos(2 \pi s),\sin(2\pi s))$ is a covering map. I tried as follows. Take $s ...
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1answer
66 views

Is Orbit Criterion an abstract nonsense? Different induced fundamental groups.

(Orbit Criterion) Let $p:\tilde X \to X$ be a covering map. If $\tilde q, \tilde q' \in \tilde X$ are two points in the same fiber $p^{-1}(q)$, there exists a covering transformation taking $\tilde ...
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131 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
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84 views

exercise 17 of hatcher page 80 chapter 1.3

Given a group $G$ and a normal subgroup $ N$, show that there exists a normal covering space $\widetilde{X} \rightarrow X $ with $\pi_{1}(X)\approx G ,\pi_{1}(\widetilde{X})\approx N $, and deck ...
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1answer
52 views

Why does the intersection change to a union in $r^{-1}(\bigcap r(V_i\cap W_i))=\bigcup V_i\cap W_i$?

Let $q: X\to Y$ and $r:Y\to Z$ be covering maps, $p=r\circ q$. If $r^{-1}(z)$ is finite for each $z$ in $Z$, $p$ is a covering map. There is a proof on ask a topologist, but I can't follow why ...
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1answer
45 views

Endowing a metric on the torus from the euclidian metric of its covering space, the plane

In Thurston's and Levy's "Three dimensional Geometry and Topology, page 6, they define the induced metric on the torus from the euclidian metric of its covering space, the plane. Specifically, for ...
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29 views

is there a specific way to find deck transformation and its related group?

is there a specific way to find deck transformation and its related group? this question came to my mind at the first time I studied deck transformation and related topics of covering space. it ...
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29 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
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1answer
18 views

Square of normal covering splits

Concerning Galois theory, let $A/B$ be a separable extension. Then $$A/B - \text{normal} \Leftrightarrow A \otimes_B A=A \oplus \cdots \oplus A,$$ where the sum has $n$ summands. Is the same correct ...
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33 views

Are two equivalent coverings of a Riemann surface also biholomorphic?

Consider a compact Riemann surface $X$. If $p_1:Y\longrightarrow X $ and $p_2:Z\longrightarrow X$ are two topological coverings of $X$, then it is univocally defined a structure of Riemann surface on ...
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43 views

using covering space technique,prove that $[G:H \cap K] \leq [G:H][G:K]$.

using covering space technique,prove that if $G$ is a group with subgroups $H$ and $K$ then $$[G:H \cap K] \leq [G:H][G:K]$$ I couldn't understand the relation between them and the covering space,so ...
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1answer
53 views

Covering of a Topological Group(Use of fundamental theorem of covering spaces)

Suppose we have two path-connected spaces $G$ and $H$. Suppose also that $G$ is a topological group with an identity element $e$ and there is a covering $$ p: H \rightarrow G $$ The problem asks that ...
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1answer
50 views

what is the Cayley complex of dihedral group $D_{4}$?

what is the Cayley complex of dihedral group $D_{4}$? I am aware of Cayley graph of $D_{4}$,can you explain to me how I should I attach 2-cell complexes to the loops to make it covering space? I ...
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3answers
54 views

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,also $A\subset X$ is a path connected subset,show that $p^{-1}(A)$ is path connected. I suppose that ...
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1answer
53 views

Covering space covers simply connected set correctly

I found this proposition in a paper stated as a well known result from topology, but I can neither find this result in my textbooks nor proof it by myself: Let $p:E \rightarrow B$ be a covering ...