Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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45 views

Lifting of a path in covering spaces. Does the lifted path lies in single sheet?

Including the proof outline for those that can't access Munkres or don't want to look. On page 342 Munkres proves a path $f:[0,1]\rightarrow B$ starting at $b_o = p(e_0)$ has a unique lifting to a ...
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Set Cover Optimization of Cubes by Balls of Equal Radius

Let $C=[a,b]\subset R^n$ be a $n$-dimensional rectangle, $a,b\in R^n$. Find the minimal radius $r$ and set $x_1, \dotsc, x_N\in R^n$, for a given integer $N$ such that $$\bigcup_{i=1}^N B_r(x_i) \...
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Discrete commutative subgroups of Euclidean isometries form lattice

I am trying to prove the following: If $g$ is a flat Riemannian metric on $\mathbb T^n$ (the $n$-dimensional torus), then $(\mathbb T^n, g)$ is isometric to a Riemannian quotient of the form $\mathbb ...
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1answer
55 views

A lift of a map from the projective plane to itself

Let g: $RP^2 \mapsto RP^2$ be a map such that $g_*$ the induced map on the fundamental group is not the zero map. We need to prove that g can be lifted to a map $T:S^2 \mapsto S^2$ such that $T(-x)= -...
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2answers
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Standard Definition of Ramified (or Branched) Cover of Topological 3-Manifolds

My work with polyhedral 3-manifolds requires me to come up with a robust definition of a ramified cover in that setting. However, I want to be sure that my definition fits into the wider scheme. ...
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1answer
43 views

Pullback of maps out of a connected & locally path connected space

I'm reading Peter May's "Concise Course in Algebraic Topology", and I'm having trouble interpreting this yellow-highlighted line. When I've seen the expression "pullback of $f$ along $g$...
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proper local homeomorphism is a covering map [closed]

I read a proof that a proper local homeomorphism is a covering map, and confused that the proof seems to be written on the assumption that the proper local homeomorphism is surjective. Why is a proper ...
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1answer
31 views

Second homology group of the universal cover of $(S^1\times S^2)\vee S^1$

This question is related to my old question: Universal cover of $(S^1\times S^2)\vee S^1$. How can we compute the second homology group $H_2(E)$, where $E$ is the universal cover of $(S^1\times S^2)\...
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How can I prove that there are $n^{n^{2}}$ maps $G\times G\to G$ where $G$ is a group of order $n$?

In the context of group theory, there's a theorem that states that for a given positive integer $n$ there exist finitely different types of groups of order $n$. Notice that the theorem doesn´t say ...
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2answers
133 views

Universal cover of figure eight is simply connected

I've been going through Viro's Topology book and I'm stuck at the universal cover of figure eight($ \mathbb{F}(a,b)$ is a free group generated by 2 elements). I can imagine the universal cover of ...
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1answer
28 views

Fundamental group of the quotient space of the $\Bbb Z$-action on $X=\Bbb C-\{0\}$ given by $n\cdot z=\lambda^nz$

Let $\lambda$ be a fixed nonzero complex number whose modulus is not $1$. Consider the $\Bbb Z$-action on $X=\Bbb C-\{0\}$ given by $n\cdot z=\lambda^nz$. I want to compute the fundamental group of ...
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1answer
31 views

Showing that a given covering is not normal

This is a question regarding exercise 5.6 of Forster's Lectures on Riemann surfaces. We have $X=\mathbb{C}\setminus\{0,1\}$, $Y=\mathbb{C}\setminus\{0,\pm i,\pm i\sqrt{2}\}$, $p\colon Y\to X$ given by ...
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2answers
153 views

Prove two equation about $H_{p} (X) $ and $kernel\ (i_{n_{*}})_{p}\ $ .

I am new with singular homology This is a fair but hard exercise for all of my classmate, we all feel dizzy after Imagine what really happen during the procedure of the question and what it really ...
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1answer
49 views

Fundamental groups of coset spaces

Let $G,H$ be topological groups. In the case that $G$ is connected and simply connected, we have $$\pi_1\left(G/H\right)\simeq\pi_0\left(H\right)\simeq H/H_0$$ where $H_0\leq H$ is the component of $H$...
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1answer
32 views

Lie Algebras of Covering of a Group is Isomorphic to the Lie Algebra of the Group.

If $\tilde{G} \to G$ is a covering of the lie group $G$, why are the associated lie algebras isomorphic? I.e., why $Lie(\tilde{G}) \cong Lie( G)$?
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1answer
42 views

The projective space $\Bbb RP^{2n}$ cannot be the total space of a nontrivial covering map

Why is the projective space $\Bbb RP^{2n}$ not the total space of a nontrivial covering map? I've heard this in class but I can't see why it holds.
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Example of non-regular Riemannian covering map

I am looking for an example of a finite Riemannian covering map $\pi: (\tilde{M}, \tilde{g}) \to (M, g)$, where $M$ is closed, which is not a regular covering map. Any hints or suggestions are very ...
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1answer
44 views

What is meant by “covering space corresponding to a subgroup of fundamental group”?

I have the following situation: $K$ is a finite simplicial complex and $\widetilde{K}\to K$ is "a covering space corresponding to the subgroup $H$ of $G=\pi_1(K)$". Can anyone say what ...
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2answers
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Hausdorff Property for a Covering Space of a Manifold $E\to M$.

I want to show that if $\pi : E \to M$ is a topological covering map and $M$ is a manifold then $E$ is a manifold. I was reading this post which helped me for the second-countability. The OP says it ...
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0answers
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On harmonic morphisms and covering maps

Let $(M^m,g)$ and $(N^n,h)$ be Riemannian manifolds. A smooth map $\pi : M \to N$ is called a harmonic morphism if for any harmonic function $f : U \to \mathbb{R}$ defined on an open set $U \subseteq ...
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1answer
63 views

Calculate $p_*(\pi_1(\tilde{X},e_i))$

Give an example of a two-fold cover $(\tilde{X},p)$ of figure eight. For those examples choose a basepoint $e$ and a base point $e_i\in \tilde{X}$ and calculate $p_*(\pi_1(\tilde{X},e_i))$ My attempt: ...
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0answers
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Given a covering map between grupoids prove that the following are the same

Let $P:E\to B$ be a cover map for connected grupoids and objects $e\in E$ and $b=P(e)\in B$, prove that the following are the same: The subgroup $P(E(e,e)\subset B(b,b)$ is normal. For every $e'\in E$...
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about covering space and compact base space

Let $p: \tilde{X} \rightarrow X$ be a covering space with $p^{-1}(x)$ finite and nonempty for all $x \in X .$ Show that $\tilde{X}$ is compact Hausdorff iff $X$ is compact Hausdorff. Proof : (1) If $\...
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0answers
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Is this induced homomorphism surjective?

Some preliminaries: Given a surface $S$ and a covering space $q: \tilde{S} \rightarrow S$, we say that $q$ is abelian if its deck transformation group is abelian. Naturally, the universal abelian ...
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1answer
52 views

Covering $\Bbb RP^\text{odd}\longrightarrow X$, what can be said about $X$?

I am looking for any argument related to the following fact, which may or may not be true. Let $f:\Bbb RP^n\longrightarrow X$ be a covering space, where $n\geq 2$. Then, $X=\Bbb RP^n$. Now, for $n=\...
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1answer
25 views

Reference request: systole of hyperbolic surface increases in covering space

I'm wondering if anyone knows of a quick proof or reference of the following fact: Let $S$ be a compact hyperbolic surface and $l > 0$. Then there exists a finite covering space $S' \rightarrow S$ ...
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2answers
29 views

For a topological manifold $X$ is it true that $X$ is a covering of $X\lor X$?

Here is my question: Let $X$ be a topological manifold. Is it true that $X$ is a covering of $X\lor X$ and $X\lor X\lor X$ and, so on. I have a intuition, $\pi_1(X\lor X)=\pi_1(X)*\pi_1(X)$.
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1answer
30 views

Isomorphic Covering Spaces without regard to basepoints and Conjugacy Classes

I have been studying Algebraic Topology from Allen Hatcher (pg.67) and I am confused with a statement in the text; first I shall give some background. We know that: If $X$ is path-connected, ...
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1answer
69 views

Using covering spaces to show that there is no continuous odd map $S^n \to S^1$

I have the following question from a past exam about basic algebraic topology: Let $p:S^1 \to \mathbb{R}\mathbb{P}^1$ be the obvious covering map. For $n\geq 2$, show that a continuous map $f:\mathbb{...
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1answer
46 views

How does Schwarz reflection principle work for a geodesic triangle inside the unit circle?

I am trying to understand section 4.7 of A course in complex analysis and Riemann surfaces by Wilhelm Schlag. The goal is to find a covering map from the unit circle to the twice punctured plane. ...
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1answer
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Lifting submanifolds

Let $\Sigma$ be a submanifold of $M$ and let $\pi\colon \widetilde{M} \rightarrow M$ be a covering map. I would like to know if it is always true that $\pi^{-1}(\Sigma)$ is a submanifold of $\...
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1answer
28 views

Inclusion of a subspace $X$ into $RP^n$ Induces Surjection on $\pi_1$

Studying for a qualifying exam, and came across this problem: Let $p: S^n \to RP^n (n \geq 2)$ be the standard two-fold covering, and let $X \subset RP^n$. Prove that $p^{-1}(X)$ is path-connected if ...
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2answers
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Is there a 3-sheeted covering map of torus S^1 × S^1? [closed]

Basically, I'm having trouble solving this question. Does anyone know how to approach this?
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1answer
45 views

Universal covering group is locally homemorphic to the group

In Knapp's book on Lie group he claims that for any separable metrizable topological group $G$ which is path-connected, locally connected and locally simply connected, the universal covering space ...
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48 views

Showing map is covering map

Consider the map $f:\mathbb{C} \setminus \{0\} \rightarrow \mathbb{C} \setminus \{0\}$ given by $$f(z)=z^n$$ for $n \in \mathbb{N}$. The book I am reading from claims that this is a covering map, but ...
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1answer
60 views

When does a universal cover exists for a Topological Space?

My question is: What are the conditions that a topological space must satisfy to admit a universal cover? And give me a counter example of a space that doesn't admit a universal cover, and which ...
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1answer
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Degree of covering map of surface and the number of sheets

Given a covering map of closed orientable surfaces $f:M\to N$, it induces a homomorphism $f_*:\mathbb{Z}\cong H_2(M)\to H_2(N)\cong\mathbb{Z}$, suppose $f_*$ is of degree $n$. Then how to see $n$ is ...
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1answer
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cover which is not Galois cover

I couldn't find any example of cover which is not Galois. The definition is following: A cover $p:Y\rightarrow X$ is said to be $Galois$ if $Y$ is connected and the induced map $\bar{p}:Aut(Y|X)\...
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1answer
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Finding subgroups of $G=\langle x,y,z~|~x^2y^2z^2 \rangle$ using covering space theory

Consider the group $G$ having a presentation $G=\langle x,y,z~|~x^2y^2z^2 \rangle$. I am trying to find all subgroups of $G$ of index 6 using covering space theory. It is well-known that the connected ...
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Galois Theory of Ramified Coverings & Classical Galois Theory

The question adresses the answer in this thread: Algebraic closure of $k((t))$ In the answer reuns used a theory relating classical Galois theory with Galois theory of ramified coverings. I'm an ...
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2answers
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Prove that the given map is not a covering map.

This problem is from Munkres' Topology section 53. I understand how p is not a covering map if U is connected. How is it not a covering map when U is not connected.
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Universal cover of disconnected Lie groups?

In fulton harris, we prove that any connected Lie group has a universal cover, but then talk about isogeny's, and the phrasing seems to suggest we're allowing disconnected ones as well. Is it true ...
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Čech cocycles and monodromy

It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
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2answers
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Transfer homomorphism in Algebraic topology

I am studying Hatcher's Algebraic topology. I am reading 3G about Transfer homomorphism. But most of the results are deemed obvious and I don't understnad why. "Let $\pi:\tilde X\to X$ be an $n$-...
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1answer
25 views

Covering maps from the torus to the torus of degree any positive integer

I'm trying to get my head around covering maps. I've seen a question that asks to show there are covering maps $p$ from the torus $T$ onto $T$ with degree equal to any positive integer. I'm not ...
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1answer
41 views

Use Seifert-van Kampen to compute the fundamental group

I am trying to generalize the problem I ask yesterday Fundamental group of sphere with antipodal points on the equator, i.e. the question is "Compute the fundamental group of the space obtained from ...
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1answer
47 views

every map $S^4\to S^2\times S^2$ has degree $0$

I am working on Algebraic Topology in the past Quals. "Prove that every map $S^4\to S^2\times S^2$ has degree $0$, i.e. the induced homomorphism in $H_4$ is $0$." Here is how I approach: the ...
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1answer
35 views

Null-homotopic covering map means the covering space is contractible

I am working on problems in the past Qual exams. "Let $f:\tilde X\to X$ be a covering map between path-connected and locally path-connected spaces. Suppose $p$ is homotopic to a constant map. ...
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1answer
31 views

unique lifing property

I'm studying Hatcher's book by myslef. I am confused about the proof of Proposition 1.34 on page 62. Given a covering space $p:\overline X → X $ and a map $f:Y → X$, if two lifts $\overline f_1 ,\...
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1answer
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Lifts of Non-Trivial Elements In The Fundamental Group In The Universal Covering

Say $p: \hat{X}\rightarrow X$ is a universal covering, $X$ is path-connected, locally path-connected, and semi-locally simply connected. Now say $\gamma\in\pi_1(X,x)$, if I can show $\gamma$ can only ...

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