For questions about or involving covering spaces.

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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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0answers
26 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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0answers
129 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
76 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
58 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
112 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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0answers
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
41 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
59 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
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1answer
86 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
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2answers
150 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
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4answers
321 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
30 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
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1answer
65 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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0answers
92 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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0answers
72 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
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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 ...
1
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1answer
38 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
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0answers
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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0answers
28 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
17 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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0answers
31 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
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1answer
42 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
49 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
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1answer
45 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
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3answers
53 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
52 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 ...
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0answers
36 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 ...
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1answer
18 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 ...
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0answers
80 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
26 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 ...
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1answer
67 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 ...
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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 ...
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2answers
50 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
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1answer
41 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
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1answer
30 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 ...
2
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1answer
117 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
56 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 ...
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1answer
124 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 ...
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0answers
131 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 ...
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0answers
49 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 ...
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0answers
63 views

Image of the map on homology induced by a covering

Let $X$ and $Y$ are two compact connected oriented 2dim smooth manifolds, and $\pi\colon X\to Y$ is an unramified covering of degree $d$. Consider the induced map $\pi_* \colon H_1 (X,\mathbb Z) \to ...
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1answer
65 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?
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0answers
22 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
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1answer
86 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 ...
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1answer
28 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: ...
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0answers
37 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 ...
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0answers
40 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
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0answers
101 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
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0answers
32 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} ...