For questions about or involving covering spaces in algebraic topology.

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1answer
19 views

Homotopy between two paths implies triviality of the loop they form

If $X$ admits a universal covering space and $\alpha$ and $\gamma$ are to homotopic paths between $x$ and $p(y)$, then $\alpha*\gamma^{-1}$ is nullhomotopic?
0
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1answer
30 views

Interpretation of points in covering spaces as homotopy classes of paths [on hold]

If $p:\widetilde{X} \to X$ is a covering map, $y \in \widetilde{X}$ determines a homotopy class of paths in $X$ joining the base point $x_0$ to the point $p(y)$. But a homotopy class of paths in $X$ ...
0
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1answer
19 views

Question About Covering Space Classification Theorem

I'm a bit confused by Hatchers choice of words here. He says "The main classification theorem for covering spaces says that by associating the subgroup $p_{*}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$ we ...
2
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1answer
28 views

Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover)

Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let $X$ be a path-connected space with $\pi := \pi_1(X,*)$ abelian and ...
2
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0answers
25 views

If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
2
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1answer
39 views

Group of deck transformations acts properly discontinuously

Let $M$ be a connected (smooth Riemannian) manifold which admits a universal cover $\tilde{M}$. Let $\Gamma$ be the group of deck transformations on $\tilde{M}$. I want to show that $\Gamma$ acts ...
0
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1answer
22 views

Let $G$ a simple connected topological group and $H$ a normal discrete subgroup, then $\pi_1(G/H,e) = H.$

I know that $G$ is a covering space for $G/H$ and there is a injection between the fundamental group of $G$ and $G/H.$ How to proceed to show that $\pi_1(G/H,e) = H?$.
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2answers
39 views

When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I ...
0
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0answers
20 views

Descending group actions to coverings

Let $X$ be a path-connected space with universal cover $\widetilde{X}$, let $Y$ be another covering of $X$ $$ \widetilde{X} \hspace{1cm} \\ \searrow \\ \downarrow\hspace{.5cm} Y\\ \hspace{.25cm}\...
2
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1answer
50 views

Does a group action lifted to the universal cover commute with the fundamental group action?

Question: Let $\varphi \colon G \to \text{Homeo}(X)$ be a group action on a topological space $X$ with basepoint $x_0$ and universal covering $\pi \colon \widetilde{X} \to X$. Then the subgroup of ...
0
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1answer
69 views

Prove that two $n$-sheeted covering space of $S^{1}$ are isomorphic.

We have a Blaschke product $B(z) \colon S^1 \to S^1$ of order $n$ and the map $f \colon S^1 \to S^1$, $f(z)=z^n$. Both maps are regular on $S^1$. We have already proved that both are $n$-sheeted ...
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0answers
40 views

Existence of the lift of a curve

Let $(M, g)$ be a complete Riemannian manifold and let $(\tilde{M}, \tilde{g})$ its universal cover. Let $\pi : \tilde{M} \to M$ be the covering map. Let $\gamma : I \to (M, g)$ be a smooth curve ...
3
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3answers
105 views

Why bother showing $S^{1}$ covers itself?

I've just been introduced to covering spaces, and one of the examples I've been shown is that $p: S^{1} \to S^{1}$, $p(z)=z^{n}$ is a covering map for every $n$. My question is: why would you care? ...
0
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1answer
41 views

Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don't come along with this, ...
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0answers
37 views

Hatcher Covering Spaces Ex. 11 & 31 and Surjectivity of the Covering Map

I am confused by the statements of a couple of the exercises in Section 1.3 of Hatcher. I think they need additional hypotheses that are not reflected in Hatcher's errata. Exercise 11: Construct ...
2
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0answers
53 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
1
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1answer
46 views

Is a Blaschke product/rational function a covering map for a $n$-sheeted covering of $S^{1}$?

We have a Blaschke product $B(z)$ of order $n$ (you can think of it as a rational function with $n$ zeros and $n$ poles), the zeros are obviously inside $\mathbb{D}$. Why is $B(z) \colon S^{1} \to S^{...
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0answers
15 views

Covering spaces of surface sphere glued to the mobius strip at one point on its boundary.

I have determined the universal covering space, but I am having trouble finding two-sheeted and three sheeted covering spaces. Any help would be greatly appreciated on how to approach this!
5
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1answer
74 views

The Euler Characteristic of $\mathbf RP^2$ is a Fraction.

Problem 22 in Section 2.2 in Hatcher's Algebraic Topology reads For $X$ a finite CW complex and $p:\tilde X\to X$ an $n$-sheeted covering space, show that $\chi(\tilde X)=n\chi(X)$. Here $\chi$ ...
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0answers
39 views

Pushforward of canonical bundle restricted to divisor isomorphic to restriction of pushfoward of canonical bundle

Consider the branched covering $f \colon X \to \mathcal{Q}_7$ of the $7$-dimensional smooth projective quadric by a smooth connected projective variety $X$. Since we have the $6$-dimensional quadric $\...
3
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1answer
67 views

Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
5
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1answer
53 views

Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
3
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1answer
37 views

Preimage of a simply closed curve under the two-dimensional antipodal map

Suppose $p:S^2\to P^2$ is the quotient antipodal map, and $J$ is a simply closed curve in $P^2$, then $p^{-1}(J)$ is either a simply closed curve in $S^2$, or two disjoint simply closed curves in $S^2$...
0
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2answers
37 views

A covering space of a Hausdorff space is Hausdorff

Let $p:Y\to X$ be a covering space. If $X$ is Hausdorff, so is $Y$. Hello, I have a question to this task. I want to show that $Y$ is a Hausdorff space. Hence for $y_1, y_2\in Y$ with $y_1\neq y_2$ ...
0
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0answers
51 views

Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
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0answers
37 views

Making a principal bundle into a covering space

Suppose $\pi : P\rightarrow M$ is a principal $G$-bundle. I want to make this into a covering map by changing the topology of $P$. By local triviality we can find for each $x\in M$ an open $U\subset ...
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0answers
29 views

Surjective group homomorphism between quotient groups

Assumptions: Assume that $G$ is a topological group and $Z_1,Z_2$ are discrete, normal subgroups of $G$ (hence central) and $G / Z_1$ and $G / Z_2$ denote the quotient groups. Assume moreover that ...
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1answer
50 views

Subgroup of Finite Index Containing a Given Finitely Generated Subgroup of a Free Group: Problem 12 in 1A in Hatcher.

Problem 1A.12 (Hatcher) Let $F$ be a finitely generated free group and $H$ be a finitely generated subgroup of $F$. Let $x\in F-H$. Show that there is a finite index subgroup $K$ of $F$ such that $H\...
0
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1answer
29 views

Covering map associated with open cover

Let $ \left\{U_i \right\}$ be an open cover of $X$. On some online sources and some MSE questions, the map $\coprod _iU_i\rightarrow X$ is given as an example for a local homeomorphism which is not a ...
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0answers
15 views

Representing Covering Spaces by Permutations: Proof Verification.

$\newcommand{\FG}{\pi_1}$ Given a covering projection $p:\tilde X\to X$, and $x_0\in X$, we can naturally define a \emph{right} action on $F=p^{-1}(x_0)$. For each point $\tilde x\in F$, and each $[\...
0
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1answer
35 views

Representing Covering Spaces by PErmutations

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher. Hatcher starts by ...
2
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1answer
43 views

What is the intuition behind covering spaces?

I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics. The definition of covering space is as ...
2
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1answer
26 views

Given that $H^1(X)=0$ on a connected space, show that all maps to $X\to S^1$ are null homotopic

Let $X$ be a path-connected, locally path-connected topological space, with $H^1(X)=0$. I would like to show that any map $f:X\to S^1$ is null homotopic, but I haven't really made any progress. ...
0
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3answers
32 views

Is a covering map on compact metric space, $k$ - to- $1$ at all points?

Let $X$,$Y$ be topological space, surjective map $\varphi:X\rightarrow Y$ is called a covering map if there is an open cover $\{U_{\alpha}\}$ of $Y$ such that for every $\alpha$, $\varphi^{-1}(U_{\...
0
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1answer
20 views

Question about covering map

Let $(X, d_1),(Y, d_2)$ be metric space, $f:X\rightarrow Y$ is called covering map, if for evry $y\in Y$, there is open set $U$ of $y$ such that $f^{-1}(U)$ is a union of disjoint open sets in $X$, ...
2
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1answer
36 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let $\phi:\...
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1answer
71 views

Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...
0
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0answers
27 views

Free action of a discrete group gives a covering space

I'd like to find a short proof of the following seemingly basic fact. Suppose a discrete group $G$ acts freely on a manifold $X$ with the quotient $X/G$ being compact. Then $X$ is a covering space of $...
2
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0answers
65 views

limit of covering spaces

Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, ...
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0answers
19 views

Deck group of a connected n-fold cover must have at most n elements

Let $p:Y\to X$ be an $n$-fold covering map, with $Y$ connected. Show that $Deck(p)$ has at most $n$ elements. My thinking was to prove this by contradiction, i.e. suppose we have distinct $\tau_1,...,...
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0answers
35 views

Fundamental group of a covering space

I understand the correspondence between the subgroups of a fundamental group $\pi_1(X)$ and the covering spaces of $X$. However, I do not understand what is implied about the fundamental groups of ...
6
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2answers
58 views

Is the Fundamental Group of space with contractible universal cover torsion free?

Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle $K$ torsion free? We have the following presentation: $$\pi_1(K) = \langle a,b: aba = b \...
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1answer
39 views

With justification, determine whether or not the following space is compact.

The space in question is the Hausdorff topological space with base β: β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} . (I have confirmed that this in fact a base of a Hausdorff ...
0
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1answer
38 views

Möbius strip covering space

how can we describe the universal covering space of the Möbius strip? the Möbius strip is a square $[0,1]\times [0,1]$ with identifications $(0,y)\sim (1,1-y)$. So my guess is that the universal ...
1
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2answers
68 views

Two lifts of a local homeomorphism

Just learning about sheaves. Suppose I have a sheaf $\mathscr{F}$ on a topological space. (The sheaf can take values in sets, let's say.) Let $E \overset{\pi}{\to}X$ be the etale space of this sheaf....
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0answers
18 views

Is this covering group $G'$ of $G$ unique?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
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0answers
34 views

what's the universal covering group of general linear group $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$

Because the general linear group is not simply connected, they must be able to be covered by some simply connected Lie group. But I cannot imagine which Lie group with same dimension can cover $GL(n,\...
1
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0answers
44 views

Fundamental group and universal cover for this quotient space

For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more ...
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2answers
49 views

Covering spaces of wedge sum of circles

Let's suppose I'd like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $ S^{1} \vee S^{2} $ it is easy to point out exactly how do all ...
0
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0answers
50 views

degree of $f\circ g$

Let $f,g : \mathbb{S}^1 \to \mathbb{S}^1$ be two continuous maps where $\mathbb{S}^1$ is the unit circle. Prove that $\deg (f \circ g) = \deg f \deg g$. I don't know at all how to do it : I first set ...