For questions about or involving covering spaces in algebraic topology.

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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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81 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
108 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
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 ...
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1answer
212 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
82 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
606 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
406 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
79 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
40 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
154 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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80 views

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
61 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
74 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
124 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
501 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
100 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
86 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
112 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
89 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
291 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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49 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
282 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
250 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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160 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
92 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
149 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
169 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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105 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
277 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
163 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
61 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
167 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
623 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
269 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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135 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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0answers
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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393 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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148 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
183 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 ...