For questions about or involving covering spaces.

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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
60 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
73 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
122 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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2answers
470 views

Function doesn't have a lift in a space related to Topologist's sine curve

I'm trying to solve exercise 1.3.7 in Hatcher's Algebraic Topology: Let $Y$ be the quasi-circle that is the union of a portion of the graph $y = \sin(1/x)$, the line segment $[-1,1]$ in the ...
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0answers
97 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
83 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
111 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
88 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
288 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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95 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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0answers
47 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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1answer
56 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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2answers
147 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
280 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
43 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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2answers
248 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
154 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
89 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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0answers
136 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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1answer
155 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 $C \backslash \{ 0,1,\lambda \}$ ...
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1answer
165 views

Can the covering transformation group of a 6-sheeted covering map have 12 elements?

Can the covering transformation group of a 6-sheeted covering map $p : X \to Y$ have exactly 12 elements? I suspect that the answer is negative, but I cannot see an invariant that shows this. We ...
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102 views

deck transformations and covering spaces

Let $p:\tilde X\rightarrow X$ be a universal covering space, and let $H\leq G$ where $G$ is the group of covering transformations. Let $q:\tilde X \rightarrow \tilde X/G$ be the quotient map which is ...
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1answer
270 views

Alternative definition of covering spaces.

in a lecture I have seen a definition of a covering space, different from what I would call the usual one (e.g. the one in Munkres): A surjective continuous map $p:E\rightarrow B$ of spaces $E$ and ...
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1answer
157 views

Connectedness implies the equinumerosity of fibers

I need to show that if $X$ is a covering space of $Y$ with the covering map $p$ and $Y$ is connected, then $p^{-1}(y)$ have the same cardinality for every $y\in Y$. I have this hint: A function ...
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1answer
60 views

fundamental group of a graph is free

Let $X$ be a connected graph, and $T$ its maximal tree. Via covering spaces and deck-transformations, how one can prove that $\pi_1(X)= \pi_1(X/T)$?
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1answer
163 views

helix and covering space of the unit circle

Does a bounded helix; for instance $\{(\cos 2\pi t, \sin 2\pi t, t); -5\leq t\leq5\}$ in $\mathbb R^3$ with the projection map $(x,y,z)\mapsto (x,y)$ form a covering space for the unit circle ...
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2answers
596 views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
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1answer
261 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
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1answer
125 views

Unit Interval is Simply Connected

Given the definition of simply connected space to be a topological space $X$ whose every connected covering is homeomorphic to $X$, i want to show that $[0,1]$ is simply connected.
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1answer
133 views

Universal covering space via fiber product

Given 3 topological spaces $X,Y,Z $ and 2 functions $ f:X \rightarrow Z $, $ g:Y \rightarrow Z$, I define the fiber product between $ X $ and $ Y $ over $ Z $ by: $$ X\times_Z Y := \{(x,y) \in X ...
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55 views

Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
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Hatcher Problem 1.3.15 [duplicate]

Would just like a sanity check. I don't see the necessity of the locally path connected condition on $A$. The proof that $\tilde{A}$ is a covering space seems straightforward. We use the ...
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1answer
82 views

Book introducing covering spaces independent of homotopy

Can anyone please suggest a book on algebraic topology which deals with covering spaces independent of homotopy, fundamental group, etc?
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2answers
199 views

A Covering Map $\mathbb{R}P^2\longrightarrow X$ is a homeomorphism

I came across the following problem: Any covering map $\mathbb{R}P^2\longrightarrow X$ is a homeomorphism. To solve the problem you can look at the composition of covering maps $$ ...
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50 views

What is a “mere cover”?

Sorry to ask such a basic question, but I'm having a lot of trouble finding a definition of this. I saw this term in Stefan Wewers' thesis and it seemed familiar, but googling "mere cover" doesn't ...
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2answers
357 views

Induced map between fundamental groups from covering map is injective

Question: Let $f : X \to Y$ be a continuous map and let $x \in X$, $y \in Y$ be such that $f(x) = y$. Then there is an induced map $f_* : \pi_1(X, x) \to \pi_1(Y, y)$ such that $f_*([\gamma]) = [f ...
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0answers
145 views

Covering space of a graph is again a graph - why??

i want to prove the statement in the heading. Thus given graph $G$ and a covering space $p:\tilde{X}\rightarrow G$ to prove that $\tilde{X}$ is also a graph. My idea was to take $\tilde{E}=p^{-1}(E)$ ...
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1answer
172 views

Is the product of covering maps a covering map?

I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times ...
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1answer
54 views

Universal covering space of connected open subset of $\mathbb R^n$

Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?
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2answers
325 views

covering space of a particular CW complex

I am trying to find all connected covers of the following space $X$ (up to isomorphisms) $X$ has one $0$-cell, two $1$-cells labeled $a$ and $b$, and three $2$-cells attached via $a^2$, $b^2$ and ...
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2answers
57 views

Show that f is covering map and find covering tranformation group

Prove that $f:\mathbb{R}^2\to T^2$ defined by $f(x,y)=(e^{2\pi i x},e^{2\pi iy})$ is covering map and also find covering tranformation group$=\{g:\mathbb{R}^2\to\mathbb{R}^2\mid g$ is diffeomorphism ...
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1answer
56 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
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Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
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1answer
42 views

What subsets of a covering space cover their image?

Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map? If it makes things easier, assume $E$ is simply connected, that is, the ...
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1answer
61 views

Finite group acting freely on Haussdorf space- Topology problem

How to prove the following problem: It is given Hausdorff space $X$ and finite group $G$ (with neutral $e$) that is acting freely on $X$. For $g\in G$, $\overline{g}:X\rightarrow X$ is ...
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2answers
124 views

How to find the induced map $f_{*} : \pi_1 (S^1 , (1,0)) \to \pi_1 (S^1 , (1,0) ) \ \ ? $

I came across this old exam question while studying for my own exam for our topology course. Let $f : S^1 \to S^1 $ be the map $z \mapsto z^n$. What is the induced map $$f_{*} : \pi_1 (S^1 , (1,0)) ...
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485 views

Algorithms for covering a rectilinear polygon using rectangles of the same size

The following is the problem description: All angles of the polygon are right. It may be convex or concave. Use rectangles of the same size to cover the polygon. The edge of the polygon and ...
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1answer
73 views

function lifting on $S^1 \times S^1$

Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 ...
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1answer
71 views

Spin group without Clifford algebras

I have to build the spin group $Spin(n)$ without use Clifford algebras. Can I find a complete description of spin group with a topological method? How can I build $Spin(n)$ as the double covering of ...