For questions about or involving covering spaces.

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12
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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 ...
0
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1answer
36 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 ...
3
votes
1answer
70 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 ...
1
vote
0answers
143 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 $S^{1} \vee S^{1}$ ...
0
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0answers
89 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 ...
1
vote
1answer
54 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 ...
1
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1answer
47 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 ...
0
votes
0answers
31 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 ...
0
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0answers
33 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 ...
1
vote
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 ...
1
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0answers
34 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 ...
0
votes
1answer
44 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 ...
1
vote
1answer
55 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 ...
1
vote
1answer
52 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 ...
0
votes
3answers
56 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 ...
0
votes
1answer
56 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 ...
5
votes
0answers
41 views

Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over ...
1
vote
1answer
20 views

Is it possible to “lower” a $\pi_1$-invariant differential function defined over the universal covering manifold to the base one?

Consider a differentiable manifold $M$ and its smooth universal covering $\pi:\tilde{M}\rightarrow M$. There is a canonical action of the fundamental group $\pi_1(M)$ on the covering manifold ...
0
votes
0answers
81 views

find a necessery and enough condition just using $\pi_{1}$.

suppose $p:\widetilde{X} \rightarrow X$ will be a covering space and $X$ is path connected and locally path connected, also $\widetilde{X}$ is connected, then find a necessary and enough condition ...
1
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1answer
28 views

“the standard two-fold branched cover of $CP^2$”

What could the following sentence mean: $\iota : S^2\times S^2 \rightarrow \mathbb{C}P^2$ is the standard two-fold branched cover, branched along the diagonal. What I can think of is to ...
0
votes
1answer
77 views

Metric completion of universal covering of punctured plane

It is known that the universal covering of the punctured plane $\mathbb C\setminus\{0\}$ is $\exp:\mathbb C\to\mathbb C\setminus\{0\}$. In real coordinates, $f=\exp:\tilde M=\mathbb R^2\to M=\mathbb ...
2
votes
0answers
24 views

Correct definition of a regular covering without global connectedness hypotheses

Let $p:Y\to X$ be a covering map of topological spaces where $X$ is assumed to be locally path connected (and hence the same is true of $Y$) but neither $X$ nor $Y$ is assumed to be connected. In this ...
1
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2answers
52 views

Is the hyperbolic plane the only simply connected hyperbolic 2-manifold?

Let $S$ be a simply connected Riemannian 2-manifold with everywhere negative curvature. Is $S$ necessarily diffeomorphic to $\mathbb{R}^2$?
0
votes
1answer
43 views

Morphism between covering spaces.

Let $p:Y\to X$ and $q:Z\to X$ be covering maps (of course $X,Y,Z$ are all Hausdorff, arcwise connected and locally arcwise connected) and $g:Y\to Z$ a morphism such that $q\circ g=p$. Then, $g$ is a ...
0
votes
1answer
36 views

Each open covering of F has a finite subcovering, how can F not complete be a problem?

I understand that if for each open covering, $\mathcal{O}$, of a set $F$, which is subset of a metric space $(X,d)$, there is a finite subcovering $\implies F$ is compact $\implies F$ is complete ...
3
votes
1answer
181 views

Universal Cover of a Surface (with Boundary)

I'm trying to see if there is a "nice-enough" way of describing/constructing the universal cover for a compact surface with n boundary components. Clearly, if $n=0$ , the classification theorem for ...
1
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1answer
64 views

Covering spaces and homotopical equivalence

I have this simple question: if $X$ and $Y$ are two topological spaces homotopically equivalents, have they the "same" covering spaces? (and if yes, in which sense?) This question derive from an ...
2
votes
1answer
190 views

Covering space(s) of $\mathbb{R}\text{P}^2$ minus one point

I know that the covering space of $\mathbb{R}P^2$ is $S^2$, and it is unique unless than isomorphism of covering spaces. Now, $S^2$ minus one point is homeomorphic to $\mathbb{R}^2$ (by stereographic ...
5
votes
0answers
152 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
1
vote
1answer
66 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 ...
0
votes
1answer
68 views

existence of double covering [duplicate]

Let $M$ be a manifold , and $\pi_1(M)=\mathbb{Z}$. then can we say, the double covering of $M$ exists and is unique?
2
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0answers
24 views

Submersions and complex structure

Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ ...
0
votes
1answer
93 views

Constructing explicit lift of a circle homeomorphism

Studying a book by Luis Barreira in the Universitext Collection -- Dynamical Systems: an Introduction -- I'm told that given $f: S^{1} \to S^{1}$ homeomorphism, it's always possible to construct a ...
1
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1answer
35 views

Finite coverings are closed.

I'm working on solving as many of the exercises in Lenstra's Galois Theory for Schemes as possible, but there is one problem I'm partially stuck on. The statement of the problem is: ...
1
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0answers
45 views

Cover polygon with rectangles

I need to cover some polygon with rectangles here's an example : The black figure in a black square is the polygon that i need to cover with those green rectangles but i need to do it more ...
2
votes
0answers
47 views

Covering Spaces and Fundamental Groups

Can somebody tell me if what I did is right? I need to Draw the based cover $\hat{B}\rightarrow B$ such that $\pi_{1}(\hat{B},v)$ corresponds to the subgroup $\langle a^{3}, a^{2}b\rangle$ ...
2
votes
0answers
112 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
2
votes
0answers
37 views

Degree and picture of a Covering map

I need to know if I am right: I need to know the degree of this covering map $R \rightarrow S$: $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ I have that genus of $R$, $g_{R} ...
1
vote
1answer
79 views

Genus of a surface

Let $S$ be a genus 3 closed orientable surface. Let $R\rightarrow S$ be a degree 2 covering map. What is the genus of $S$ ? Do I have to use the Euler characteristic of a surface presentation which is ...
1
vote
2answers
127 views

Covering map of a Torus

How would I draw (describe) a covering map given by $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ $T^{2}\#T^{2}\#T^{2}\#T^{2}\rightarrow T^{2}\#T^{2}$ and what would be the ...
2
votes
1answer
83 views

Find The Automorphism Group

I am taking my first course in Geometry and Topology and we are seeing the automorphism group of a covering. In class, my teacher gave some graphs and their automorphism groups, but he did not explain ...
1
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1answer
73 views

Automorphism groups of Graphs

What are the automorphism groups of the following regular covering spaces? I think the first picture is an 8-degree cover of the figure 8, whereas the second one is an infinite degree cover. I ...
1
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0answers
25 views

Covering spaces and Automorphisms

I need to find for the groups $G$ a connected degree-4 cover $\hat{B}\rightarrow B$ such that Aut($\hat{B}\rightarrow B$) is isomorphic to $G$ $G \cong 1$ $G \cong \mathbb{Z}_{2}$ $G \cong ...
1
vote
1answer
71 views

Connected Covering Space of a Bouquet of 3 Circles

Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. What is $\pi_{1}(\hat{X})$ (It is a free group of what rank?). Any hints?
0
votes
1answer
79 views

Universal covering Spaces Drawings

I just have trouble drawing universal covers, how can I draw the universal covers of the following spaces: $X$ is the union of a circle with a projective plane $\mathbb{P}^2$ identified along a ...
3
votes
1answer
278 views

Besicovitch Covering Lemma

We just finished our unit on covering lemma's in my analysis class and my professor proved both the Vitali and Besicovitch covering lemma's (for finite and infinite coverings) using balls. He ...
1
vote
1answer
100 views

Homotopy groups of a covering space

This is a question related to the exercise 2218 from the book "Problems and Solutions in Mathematics" by Ta-Tsien, $2^{nd}$ Ed. Let $Z$ denote the figure 8 space, $Z = X \vee Y$, $X$ and $Y$ circles. ...
3
votes
2answers
80 views

For any element $e$ of an open set $V$ of a covering space, does there exist a sheet $S$ such that $e\in S\subseteq V$

Let $p:E\rightarrow X$ be a covering map. Let $V$ be any open subset of $E$ and $e$ be any element of $V$. I feel that the following statement must be true: There exists an evenly covered open subset ...
2
votes
1answer
169 views

On double covering of projective plane and map preserving antipodal points

Let $p:S^2\to P^2$ be the double covering of the real projective plane. Let $g:P^2 \to P^2$ be a map such that its induced homomorphism on fundamental group is not trivial. I'd like to show that ...
1
vote
1answer
78 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...