For questions about or involving covering spaces.

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2
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1answer
38 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
48 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
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0answers
51 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 ...
1
vote
1answer
63 views

Is a covering space of a completely regular space also completely regular

I'm trying to solve a problem in Munkres' Topology book. Let $p: E \rightarrow B$ be a covering map and suppose that $B$ is completely regular (for any closed subset $A$ and disjoint point $a$ there ...
2
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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
39 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?
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) ...
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 ...
1
vote
1answer
376 views

Klein bottle covered by the torus

Maybe this is an idiot question and I'm missing something very trivial. This question question was asked here before, but the answer (which apparently is equal to the one that I created) seems ...
0
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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 ...
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 ...
1
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0answers
31 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. ...
4
votes
1answer
64 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 ...
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
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
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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 = ...
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 ...
0
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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
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 ...
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 ...
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
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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
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2answers
140 views

If a connected open set is evenly covered, then its preimage is uniquely partitioned into slices

This is from Topology by Munkres: Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of ...
2
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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
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
28 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 ...
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
117 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
53 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 $d$ ...
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 ...
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 ...
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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 ...
7
votes
1answer
141 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 ...
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 ...
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
3answers
324 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
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 ...