For questions about or involving covering spaces.

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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
151 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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138 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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59 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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64 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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66 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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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 $X$ ...
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90 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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31 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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38 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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41 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 $\langle a^{3}, a^{2}b\rangle$ ...
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103 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 ...
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33 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} ...
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1answer
76 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 ...
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2answers
110 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 ...
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1answer
82 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 ...
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1answer
69 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 ...
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24 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 ...
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1answer
64 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?
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1answer
72 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 ...
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1answer
255 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 ...
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1answer
95 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. ...
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2answers
77 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 ...
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1answer
144 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 ...
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1answer
77 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 ...
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1answer
309 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 ...
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3answers
296 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 ...
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1answer
68 views

What's the colimit of the n-sheet covering spaces over the circle?

I was thinking in computing the filtered colimit of the n-sheet covering spaces $f_n: \mathbb{S}^1 \longrightarrow \mathbb{S}^1$ ($f(z) = z^n$) in the comma category of topological spaces with the ...
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0answers
39 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
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2answers
123 views

How to compute the fundamental group of a necklace of $\mathbb{S}^1$' s?

I was trying to compute $\pi_1 (X)$ where $X =$ "necklace of $n$ $\mathbb{S}^1$'s". At first, I tried using Van Kampen theorem however I could not find open sets $U$ and $V$ such that $U \cap V$ is ...
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finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
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1answer
56 views

Monomorphisms and epimorphisms in the category of finite coverings of a topological space

I'm working my way through Lenstra's Galois Theory for Schemes, and I've run into a bit of a conundrum with Exercise 3.14(b). In this exercise, we consider the category $\textbf{FC}_X$ of finite ...
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1answer
64 views

Any homeomorphism is a covering map

Prove that any homeomorphism is a covering map. My thought: Let $p:X\to Y$ be a homeomorphism. Choose $y\in Y$. Then $Y$ is a open neighbourhood of $y$. Since $p$ is a homeomorphism, $p^{-1}(Y)=X$ ...
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3answers
103 views

Covering Space of the Pearl Necklace

Let $S^{2}_{1},\dots, S^{2}_{n}$ be disjoint copies of the unit sphere, and, for each $i\in\{1,\dots, n\}$, let $p_i,q_i\in S^{2}_{i}$ be distinct points. Define an equivalence relation $E\subseteq ...
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92 views

Connected Components of Covering Space

Let $\pi : \tilde M\to M$ be a covering map, $K_1, K_2\subset\tilde M$ are two different connected components and there exists such points $x\in K_1, y\in K_2$ that $\pi(x)=\pi(y)$. In other words, ...
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1answer
71 views

Lifting elements of $SO(3)$ to $SU(2)$.

Let $A$ an element of ortogonal group $SO(3)$ such that the orders of $A$ is $>2$. We have that $SU(2)$ is a $2$-fold cover of $SO(3)$: $$ \mathbb{Z}_2 \to SU(2) \to SO(3) .$$ So how can I build a ...
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1answer
96 views

Does every map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n$ lift to a pair of maps $S^n\rightarrow S^n$?

Question: Given a continuous map $f:\mathbb{R}P^n\rightarrow\mathbb{R}P^n$, is there automatically a continuous map $g:S^n\rightarrow S^n$ such that $f,g$ commute with the covering map ...
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1answer
84 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
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3answers
276 views

Existence of a Minimal Cover

I'm well aware that for the sequence $x_n=\frac{1}{n}$, $\text{inf }x_n=0$ but $0 \notin (x_n)$. This made me think about something similar but when we are no longer thinking about existence of a ...
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0answers
88 views

Fundamental group and path-connected

Let $p:E \rightarrow B$ be a covering space, $E$ and $B$ are path-connected. Let $A$ be a path-connected subset of $B$ . How to use fundamental group to give a sufficient and necessary condition to ...
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42 views

Group of covering transformations

The group of automorphisms of a covering $p: E \mapsto X$, to be denoted $Aut(E,p)$, is usually referred to as the group of covering transformations. If $p: E_1 \mapsto E_2$ is an isomorphism of ...
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50 views

Example of a Spread which is not Complete

This is a continuation of an original question about spreads, which are something like pre-branched covering spaces. See the basic definitions here: A Complete Spread I have an example of a spread ...
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132 views

Covering map + homotopy equivalence = homeomorphism?

How to show that a covering map which is also a homotopy equivalence is a homeomorphism?
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2answers
276 views

If $\|\left(f'(x)\right)^{-1}\|\le 1 \Longrightarrow$ $f$ is an diffeomorphism

Let $f:\mathbb{R}^n \longrightarrow \mathbb{R}^n,f\in C^1(\mathbb{R}^n)$ such that $\forall x \in \mathbb{R}^n\;,\;f'(x)$ is an isomorphism and: $$ \|\left(f'(x)\right)^{-1}\|\le 1\;,\forall x \in ...
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1answer
37 views

A Complete Spread

I am reading R. Fox's "Covering Spaces with Singularities", which deals with a careful definition of branched covering spaces. I am having trouble understanding the exact definition and importance of ...
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220 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
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1answer
131 views

Prove that this covering map is a homeomorphism

Let $p \colon E \to X$ be a covering map. Let $s \colon X \to E$ be continuous. If $p \circ s = \operatorname{id}_{X}$, show that $p$ is a homeomorphism. We know that $p$ is a continuous surjection. ...
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1answer
78 views

Can I decompose a compact set in a finite number of convex set?

My problem is in a finite-dimensional space. I look at $\mathcal{X}$ the support of a function $f$, that is continuous and has bounded support. \begin{eqnarray} \mathcal{X}_o & = & \{x \in ...
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107 views

find the connected covering space of $\mathbb{R}P^2 \lor \mathbb{R}P^2$

This is a problem on Hatcher' book. How to find all the connected covering spaces of $\mathbb{R}P^2 \lor \mathbb{R}P^2$? I don't know where to start. Is there a general way to construct the ...
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153 views

Covering space of $C \backslash \{0,1,\lambda \}$

Let $\lambda\in C \backslash \{0,1\}$, $E= \{(x,y) \in C^2 : y^2=x(x-1)(x-\lambda),\ x\neq 0,1,\lambda \}$. Prove that $E$ is a connected $2$-fold covering space of $C \backslash \{ 0,1,\lambda \}$ ...