For questions about or involving covering spaces in algebraic topology.

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2
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1answer
288 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 ...
1
vote
1answer
167 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 ...
2
votes
1answer
62 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 $\pi_1(X)= \pi_1(X/T)$?
1
vote
1answer
176 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 ...
5
votes
3answers
687 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 ...
8
votes
1answer
277 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 ...
-3
votes
1answer
126 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.
1
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1answer
138 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 ...
2
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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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0answers
78 views

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 ...
1
vote
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?
4
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2answers
200 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 $$ ...
0
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0answers
51 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 ...
4
votes
2answers
414 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 ...
0
votes
0answers
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 $\tilde{E}=p^{-1}(E)$ ...
2
votes
1answer
187 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 ...
3
votes
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 $\mathbb{R}^n$?
3
votes
2answers
342 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 ...
1
vote
2answers
63 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 ...
0
votes
1answer
58 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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0answers
88 views

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 $\gcd(n,m)$ ...
1
vote
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 ...
1
vote
1answer
63 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 ...
0
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2answers
126 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)) ...
3
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0answers
509 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 ...
0
votes
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 ...
2
votes
1answer
73 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 ...
1
vote
1answer
76 views

Finding sheet number of torus using universal cover

I have a question from my lecture notes that I need clearing up: Given a covering $p: S^1\times S^1 \rightarrow S^1\times S^1$ by $p(z,w)=(z^a w^b,z^c w^d), a,b,c,d\in\mathbb{Z}$, we want to find the ...
7
votes
2answers
783 views

Covering space Hausdorff implies base space Hausdorff

There is an exercise problem in Hatcher's Algebraic Topology book asking to show that if $p:\tilde{X}\rightarrow X$ is a covering space with $p^{-1}(x)$ finite and nonempty for all $x\in X$, then ...
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0answers
50 views

The covering space of a region contained in complex plane delete two points.

We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
3
votes
1answer
362 views

The restriction of a covering map on the connected component of its definition domain

Suppose $p:Y\to X$ is a covering map, $X,Y$ are manifolds and $X$ is connected. If $Z$ is a connected component of $Y$, I wonder if the restriction of $p$ on $Z$ is also a covering map? If not, what ...
2
votes
2answers
228 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
6
votes
1answer
387 views

Induced map on homology from a covering space isomorphism

Suppose $S^1 \times \mathbb{R}P^2$ covers some space. Why is it that any covering space isomorphism $h$ induces the identity map on $H_1$? I don't see how to prove this except maybe from looking at ...
3
votes
1answer
78 views

Is $f$ necessarily a covering?

Let $f : X \rightarrow Y$ be a continuous map of spaces where $X$ is compact Hausdorff , $Y$ is Hausdorff and both spaces are path-connected and locally path-connected. Suppose that for every $x \in ...
4
votes
1answer
121 views

Restriction of Covering Space

I'm studying for an exam, and got stuck on the following exercise: Find all two-sheeted covering spaces for $X =\mathbb{S}^1 \vee \mathbb{S}^1$. Label the two circles of $X$ by $a$ and $b$. Attach ...
6
votes
1answer
375 views

The action of the group of deck transformation on the higher homotopy groups

This is for homework. I'm supposed to do exercise 4.1.4 in Hatchers "Algebraic Topology", which is to show that given a universal covering $p: \tilde{X} \to X$ of a path-connected space $X$, the ...
0
votes
0answers
64 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
2
votes
1answer
118 views

All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
7
votes
2answers
508 views

A locally constant sheaf on a locally connected space is a covering space; Proof?

As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
1
vote
1answer
62 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
0
votes
1answer
58 views

Plane rotation homomorphism

I want to show that the following set (given group structure with matrix multiplication) $$D=\left\{ D(\theta)=\begin{pmatrix} e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2} \end{pmatrix}, 0 \leq ...
2
votes
3answers
142 views

Relationship between the fundamental group and the natural equivalence classes of its universal cover

For a universal covering $p: Y \to X$, under the equivalence relation $y_1 \sim y_2$ if $p(y_1) = p(y_2)$, $Y$ admits the quotient map $\, \, \, q: Y \to Y / \sim$. There is a natural bijection $\bar ...
2
votes
1answer
68 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
4
votes
2answers
128 views

Extending a quotient map to a covering map on $\mathbb{RP}^2$

Why can we not extend the quotient map $q:[0,1]\times[0,1] \to \mathbb{RP}^2$ to a covering map, $\mathbb{R}^2 \to \mathbb{RP}^2$?
3
votes
2answers
394 views

Universal cover of a figure eight?

An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'. Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can ...
2
votes
1answer
151 views

Universal cover modulo the monodromy action

Let $\tilde{X} \to X$ be the universal cover of a connected, locally path-connected and semi-locally simply connected topological space $X$. Is it always true that the orbit space $$ \tilde{X} \;/\; ...
2
votes
2answers
205 views

Is the universal covering surface orientable?

Let $M$ be a smooth, say also closed (compact and without boundary) surface. Is it true that its universal covering surface is orientable?
3
votes
1answer
166 views

Domain is Hausdorff if image of covering map is Hausdorff

Suppose that $p:X\rightarrow Y$ is a covering map. Show that if $Y$ is Hausdorff, then so is $X$. I have an answer but I'm not sure if it's right? By definition of Hausdorff, $\forall x,y, \in Y, ...
2
votes
0answers
231 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then ...
2
votes
1answer
379 views

Domain is locally path-connected if image of a covering map is locally path-connected.

Show that if $p:X\rightarrow Y$ is a covering map and $Y$ is locally path-connected, then so is $X$. How do you go about proving this? I can think of two ways of doing this, either by definition of ...