For questions about or involving covering spaces.

learn more… | top users | synonyms

1
vote
0answers
28 views

Unique Path lifting of covering map

Let $p:E\rightarrow B$ be a covering map (in particular $p$ is a fiber bundle with discrete fiber). We want to prove the following: Given a commuting diagram of the following form: $\{0\}\rightarrow ...
1
vote
2answers
35 views

Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...
3
votes
0answers
45 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
2
votes
1answer
40 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
2
votes
1answer
50 views

Kernel of induced map between singular chain groups

Let $p : \widetilde X \to X$ be a two-sheeted cover. This induces $p_\sharp : C_n(\widetilde X; \mathbb Z_2) \to C_n(X; \mathbb Z_2)$. I can show that $p_\sharp$ is surjective by noting that every ...
2
votes
0answers
52 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
0
votes
1answer
45 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
2
votes
0answers
30 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
0
votes
1answer
40 views

How to build a covering space ?? [closed]

I have to build a covering space $ p:\mathbb{R} ^{2} \rightarrow S^{1} \times S^{1}$ How to do it?
1
vote
1answer
27 views

Show that the free group on $n$ generators is a finite index subgroup of $F_2$

Using covering spaces, prove that for each integer $n \geq 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators. I get how the cayley graph of $F_n$ would be ...
3
votes
1answer
22 views

Based covering maps for a bouquet of two circles

For each of the following subgroups of $$ \left \langle x,y \right \rangle = \pi_{1}(S^{1}\vee S^{1}) $$ construct a based covering map $$ \ p:(\tilde{X},\tilde{b})\rightarrow (S^{1}\vee S^{1},b) ...
0
votes
2answers
29 views

Showing a topological space covered by connected subspaces is connected

'Let $X$ be a topological space and let $(U_i)_{i \in I}$ be a cover of $X$ by connected subspaces $U_i$. Supposed for all $i,j \in I$ there exists some $n \geq 0$ and $k_0,...,k_n \in I$ such that ...
1
vote
0answers
33 views

Uniqueness of the universal covering space (up to an isomorphism)

Let $Y_1$, $Y_2$ be universal covering spaces of some topological space $X$. I want to show that $Y_1$ are $Y_2$ are isomorphic. Denote $p_1 \colon Y_1 \to X$, $p_2 \colon Y_2 \to X$ the projections. ...
0
votes
1answer
44 views

For every connected space X and an open cover U, every two points has a simple chain containing them

I am trying to prove this theorem saying: " A space X is connected, if and only if for an open cover U of X, every two points in X has a chain between them". I cant prove only if part (a connected ...
4
votes
1answer
65 views

Covering of a CW-complex is a CW-complex

Let $X$ be a CW- complex, with filtration $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X$. Let $p\colon E \to X$ be a covering space. Prove that $E$ is a CW complex with filtration ...
0
votes
0answers
13 views

Homotopy Lifting Property for Relative Homotopies

If $p:\tilde X \to X$ is a covering space, the homotopy lifting property says that if $f_t$ is a homotopy in $X$ and $\bar{f_0}$ a lift of $f_0$ then there exists a unique lift $\tilde{f_t}$ of $f_t$ ...
1
vote
1answer
36 views

Use of Banach-like Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified.
0
votes
0answers
17 views

About Galois Covering Theory

so I am studying somethings about Galois Covering and I am writing a beamer to present for my friends of the university. But I would like of somethings about the author of Covering Galois Theory to ...
0
votes
1answer
28 views

Covering associated to a map

I'm stuck with this exercise. Let $$p : E \to X $$ be a covering map. Y is a connected and locally path-connected topological space, $$f : Y \to X $$ is a continous map. The claim is that $$f^*p ...
1
vote
1answer
74 views

Infinite degree covering space of a bouquet of circles

I am having a hard time showing that every finite group is the automorphism group of some infinite degree covering space of a bouquet of circles (rose). Here's what I have done so far: Let $G = ...
2
votes
1answer
83 views

Any finite morphism to $\mathbb P^2$ is ramified

I want to prove that $\mathbb P_k^2$ is 'etale simply connected or that every finite morphism $X \to \mathbb P_k^2$ is ramified. Firstly I assume $X$ is regular. So if $X \to \mathbb P_k^2$ is ...
0
votes
1answer
29 views

Question about covering spaces extending inverse.

If $p$ is a cover map how would I be able to show that $x\rightarrow p^{-1}(x)$ extends to a functor $p^{-1}$ originating from the Fundamental Group of $X$?
1
vote
1answer
14 views

Correctness of reasoning about finiteness of degree of a covering map

Let $q$ be a covering $ q \colon \mathbb{R} P^{2n} \to X$, where $X$ is path-connected. Call $V_x$ the open nbhd of $x \in X$ given by the definition of covering map. We first note that $X$ must be ...
1
vote
2answers
42 views

Possible degree of a cover $p \colon S^{2n} \to X$

I'm asked to compute all the possible degrees of a covering space $S^{2n} \to X$, where $X$ is a path connected space. My idea is to try to show that these degrees can only be $1$ (take the identity ...
1
vote
2answers
55 views

Why existence of universal covering implies that the base space be locally path connected?

I am reading Chapter 13, the chapter about classification of covering spaces, of J.Munkres' Topology. My confusion raised when I read Corollary 82.2. which says: the space $B$ has a universal ...
3
votes
1answer
34 views

What is the $\pi_1$-action on the hom-sheaf between two finite etale covers?

Say you have two finite etale covers $X\rightarrow S$, $Y\rightarrow S$. The hom sheaf $\mathcal{H}om_S(X,Y)$ on the etale site $\text{Sch}/S$ is finite locally constant, hence representable by some ...
1
vote
1answer
52 views

Explicitly building Top. space with trivial homology group and non trivial fundamental group, w/o CW-complexes

I know this is a pretty famous question here, but I was asked to show explicitly such space, during a bachelor lecture, without using any CW-complex result. I started working using some covering ...
2
votes
3answers
97 views

Is a path connected covering space of a path connected space always surjective?

If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where ...
1
vote
0answers
27 views

Covering a finite collection intervals

I was trying to solve one of the problems in stein's real analysis book Suppose $I_1, I_2, . . . , I_N$ is a given finite collection of open intervals in R. Then there are two finite sub-collections ...
1
vote
1answer
39 views

Perfect Coverings

This is a problem from Brualdi and no solution is given for this. The Question goes as .... Let g(n) be the number of different perfect covers of a 3-by-n chessboard by dominoes. Evaluate g(6). I ...
1
vote
2answers
50 views

Cover of a finitely punctured plane

Let $X_n$ be the plane with a finite number $n$ of punctures, and let $p : Y \rightarrow X_n$ be a covering map (it may have infinite degree). Can we say anything about the topology of $Y$? (I know ...
0
votes
1answer
29 views

Question about Lebesgue Covering Dimension

Suppose we have a metric space equipped with two different metrics: $(X,d), (X, d')$. What must be true of the metrics: $d, d'$ in order for $X$ to have the same Lebesgue covering dimension? A ...
2
votes
0answers
60 views

Branched coverings of unit disk

Is there a classification of branched coverings of the closed unit disk $\mathbb{D} =\{z\in \mathbb{C} \ | \ |z| \leq 1 \}$? Here we consider only branched covering projections which restrict to ...
1
vote
2answers
80 views

Rigorous Covering Space Construction

Construct a simply connected covering space of the space $X \subset \mathbb{R}^3$ that is the union of a sphere and diameter. Okay, let's pretend for a moment that I've shown, using van Kampen's ...
0
votes
1answer
122 views

a covering map is open?

$E,B$ are topological spaces and lets say that $p:E\to B$ is a covering map. $p$ is open? i tried to show it as follows: let $U$ be an open set in $E$, and now for every $x\in p(U)$, $p(x)\in B$ ...
1
vote
1answer
54 views

Construction of a covering space as a fibre bundle

In a direct proof of the equivalence of categories between the covering maps $p:(\hat X, \hat x) \rightarrow (X,x)$ of a topological space $(X,x)$ for sufficiently beautiful $X$ and the ...
0
votes
1answer
57 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
votes
1answer
37 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
votes
1answer
111 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
votes
0answers
76 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 ...
1
vote
0answers
54 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
80 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
votes
1answer
60 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
votes
1answer
142 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
votes
2answers
43 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
votes
2answers
82 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
votes
0answers
41 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
votes
1answer
55 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
vote
1answer
66 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 ...