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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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Smoothness of the action of the deck transformation group

In page 163 of John M Lee's "Introduction to Smooth Manifolds" (second edition), for a given smooth covering map $\pi:E\to M$, when the author proves the smoothness of the action of the deck ...
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How to prove the continuity of the path lifting function of covering spaces.

Let $p\colon E \to B$ be a covering map. The path lifting function $\varphi\colon E \times_p B^I \to E^I$, where $E \times_p B^I := \{(e, \gamma) \in E \times B^I : \gamma(0) = p(e)\}$ is a pullback ...
HWC's user avatar
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covering space action implying a covering map [closed]

I am tried to prove the following claim: Let $X\circlearrowleft G$ be a covering space action, $H<G$ a subgroup. Then the projection $p:X/H\rightarrow X/G$ is a covering map. $x\underset{G}{\sim} ...
mappingmoe's user avatar
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Manifolds and covering maps [duplicate]

I am studying topological manifolds; I know that if $M$ is an $m$-topological manifold and $p:M\rightarrow N$ is a covering map, then $N$ is an $m$-manifold. I know how to prove the existence of local ...
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Universal Covering Space of an $n$-dimensional CW Complex

Let $X$ be an $n$-dimensional CW-complex. Let $\tilde{X}$ be its universal covering space. I want to determine if $$H_i(\tilde{X},\mathbb{Z})=0,\,\,\,\,\, i\geq n+1.$$ I would like to say that $\tilde{...
Akhalbing's user avatar
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Locally compact Hausdorff covering space

I am trying to prove the following conjecture: Let $\pi : E\to B$ be a covering map. If $E$ is locally compact Hausdorff, then so is $B$. (The converse is known to be true, cf. Exercise 6 in Section ...
Nick F's user avatar
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Any smooth mapping from $\mathbb{R}^n$ into $S^1$ is of the form $e^{if(x)}$?

Let $F\colon \mathbb{R}^n \to S^1$ be a smooth mapping. Then, I strongly suspect that there must be a smooth function $f\colon \mathbb{R}^n \to \mathbb{R}$ such that \begin{equation} F(x) = \exp \big( ...
Keith's user avatar
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Proving $\Theta \colon \mathbb{R}^{2} \rightarrow T$ is covering map and image of $\Phi$ is homeomorphic to Mobius strip

I came across the following problem. Let $R > r > 0$, consider the Torus imbedded in $\mathbb{R}^{3}$ as $T = \{(x, y, z) \in \mathbb{R}^{3} \mid (\sqrt{x^{2} + y^{2}} - R)^{2} + z^{2} = r^{2}\}...
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Prove $p \colon \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} \colon z \mapsto \bar{z}^{2}$ is a covering map

I'm stuck on this problem and generally struggle to show that some maps cover maps. Consider $p \colon \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} \colon z \mapsto \bar{z}^{2}$. Prove that $p$ is a ...
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Surjective, closed local homeomorphism with finite fibers is a covering map?

Let $E,X$ be connected, locally path connected topological spaces and $\pi: E \to X$ a surjective, closed local homeomorphism with finite fibers. Is $\pi$ necessarily a covering map? I know that this ...
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Surjective local diffeomorphism with finite fibers is a covering map? [duplicate]

Let $E, M$ be connected manifolds and $\pi: E \to M$ a surjective local diffeomorphism with finite fibers. Is $\pi$ necessarily a covering map? I know that this is true if $E$ is compact, or more ...
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What action on $\mathbb{R}^2$ yields a closed genus 3 surface

The closed, connected, orientable surface of genus 3 has universal cover $\mathbb{R}^2 \rightarrow \Sigma_3$. As such, $\Sigma_3$ can be described as a quotient of $\mathbb{R}^2$ by some properly ...
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Holomorphic map with no singular values is a covering map?

I am working on Problem 8-c from Milnor's Dynamics in One Complex Variable, which describes a necessary and sufficient condition for a holomorphic map $f : S \to S'$ between Riemann surfaces to be a ...
Nick F's user avatar
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Finite covering of Hirzebruch surfaces

Question Let $F_n = \mathbb{P}(\mathscr{O}_{\mathbb{P}_1}\oplus\mathscr{O}_{\mathbb{P}_1}(n))$ be an Hirzebruch surface and consider its finite covering(or double covering) $f: X \to F_n$. Let $f$ be ...
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Question about covering space of punctured unit disk

Let $\mathbb{H}$ be the upper half-plane and $a \in \mathbb{R}^*,\, p:\mathbb{H} \to \mathbb{D} \setminus \{0\}, \, p(z) = e^\frac{2 \pi i z}{|a|}$. I want to show that $p$ is a covering map. What I'...
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Why can we choose lifts $\tilde{\alpha}$ and $\tilde{\beta}$ of $\alpha$ and $\beta$ that have the same endpoints in $\partial\mathbb{H}^2$?

My question refers to the proof given in http://euclid.nmu.edu/~joshthom/Teaching/MA589/farbmarg.pdf, page 34. Proposition 1.10 states the following: Let $\alpha$ and $\beta$ be two essential simple ...
olchew's user avatar
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Munkres Topology Section 53 Exercise 6

$\Large\textbf{Motivation}$ At the time of writing this, I am studying algebraic topology wanted to get more familiar with covering spaces. In doing so, I went through some exercises in Munkres' ...
C Squared's user avatar
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Example of continuous function without lifting

I have seen the following theorem in class: Theorem: Let $\rho: \hat{X}\rightarrow X$ be a covering map, $x_0\in X, \ \hat{x}_0\in \rho^{-1}(x_0)$, $f:Y\rightarrow X$ continuous with $ y_0\in Y$ ...
Valere's user avatar
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Confusion about normal covering spaces

I struggle with understanding and using the following definition. A covering space $p:\tilde{X} \to X$ is called normal if for each $x \in X$ and each pair of lifts $\tilde{x}$,$\tilde{x}'$ of x there ...
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Existence of fundamental domain

I am trying to prove the existence of a fundamental domain for the free and properly discontinuous action of the group of deck transformations $\Gamma = \mathrm{Deck}(X)$ of the universal cover $\pi : ...
fresh's user avatar
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Construct covering space with given fiber

Suppose $(X, x_0)$ has a universal cover and $A$ is a left $\pi$-set. How do I find a covering map $q: Y \to X$ with the fiber of $x_0$ isomorphic to $A$ as a $\pi$ set? I vaguely know that this has ...
Nancium's user avatar
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covering space of torus [closed]

What is the covering space of the torus $T = S^1 \times S^1 $ corresponding to the subgroup $2\mathbb {Z} \times 2\mathbb {Z} $ of $ \mathbb {Z} \times \mathbb {Z} $? Here is my explanation, is it ...
User1's user avatar
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Orientation covering

My question is about whether or not the image of an open set is still an open set. I'm going to write the construction of the orientation covering and then I'll ask what I can't figure out. Let $M$ be ...
Gabriele's user avatar
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Confusion about finding a covering space for $\langle a^2, (ab)^4, b^2\rangle \leq \pi_1 (\mathbb S ^ 1 \vee \mathbb S ^ 1)$ (Hatcher 1.3.12)

I am confused about the covering space of the wedge sum of two circles that corresponds to a subgroup $\langle a^2, (ab)^4, b^2\rangle$ (reminiscent of $D_4$) of the fundamental group of the base ...
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Covering properties of non-constant holomorphic function $f: X \rightarrow \mathbb{C}$

I'm working through a proof that Riemann surfaces are second countable, and one of the main steps is showing that if $X$ is a connected Riemann surface such that there is a non-constant holomorphic ...
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Fundamental group of covering space as the kernel of homomorphism

Consider a surjective homomorphism $\theta: Z_2 * Z_3 \to S_3$ ($S_3$ is the symmetric group on 3 objects) given by mapping the generators to elements of order 2 and 3 in $S_3$ respectively. By ...
ricci_borel's user avatar
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1 answer
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Bijection involving the fundamental groupoid of a manifold

Let $M$ be a smooth manifold. I read in this post, that there is a bijection between the fundamental groupoid $\Pi(M)$ and $(\tilde{M}\times \tilde{M})/\pi_1(M)$, where $\tilde{M}$ is the universal ...
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Covering Map Associated to Map from Homology

Let $X_n$ be a connected $2$-dimensional manifold with nonempty boundary and set $X_n^*:=X_n-\{x_0\}$. Assume that $H^1(X^*_n;\mathbb{R})\cong \mathbb{R}$. Then since $$H^1(X_n^*;\mathbb{R}) = \text{...
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Calculating $p_{*}(\pi_1(E, e_0))$ for a pentagonal cover of $\mathbb{S}^1 \vee \mathbb{S}^1$

I am studying covering spaces as part of my topology course and designed the following problem as a means to assess my current understanding of the topic. I don't know how to include such a ...
Talmsmen's user avatar
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Fundamental group of $S^n$

I begin to learn algebraic topology from J. R. Munkres. Topology (Second Edition) (Upper Saddle River: Prentice Hall, 2000) There is generel theorem for calculating fundamental groups of some spaces ...
Nilufar Rajabova's user avatar
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Generators for a Covering Space of $\mathbb{S}^1 \vee \mathbb{S}^1$. Are they $\langle a,b^2,bab\rangle$ or $\langle a, b^2, bab^{-1}\rangle$?

I have a short question regarding a covering space of $\mathbb{S}^1 \vee \mathbb{S}^1$. My instructor stated that the generators were $\langle a,b^2,bab\rangle$ instead of $\langle a, b^2, bab^{-1}\...
Talmsmen's user avatar
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1 answer
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Automorphisms of non normal covering

It is known that if $p:E\to X$ is a regular covering, then by defining its deck group $$Aut(E,p):=\{\varphi:E\to E\ \ homeo\ \ s.t.\ p\circ\varphi=p\}$$ We obtain an isomorphism $\frac{\pi_1(X,x)}{p_*(...
Gabriele's user avatar
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Is there anyway to guarantee probability mass coverage?

If I have a probability density function on $\mathbb{R}^n$. I can sample $m$ points from it. Is there anyway to get an estimate of how much probability mass is covered by balls of radius $\delta$ ...
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Describe the monodromy of a holomorphic map $f:\mathbb{P}^1\to \mathbb{P}^1$

I am currently studying monodromy from Rick Miranda's book "Algebraic Curves and Riemann Surfaces" and I have the following problem: Let $f:\mathbb{P}^1\to \mathbb{P}^1$ be defined via the ...
Nerhú's user avatar
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Loop lifts uniquely to another loop

Question: Let $B$ be a space with fundamental group $\pi_{1}(B, b)$, and let $E$ be a covering space of $B$ with covering map $p$. Let $f$ be a loop at $b$ with $[f]$ the identity in $\pi_{1}(B, b)$. ...
ByteBlitzer's user avatar
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2 answers
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On a definition of covering space in terms of fiber bundles

I'm currently taking a course in (smooth) fiber bundles. I am not very used to covering spaces, but I'm trying to understand them in the language of fiber bundles. I would like to know whether the ...
Níckolas Alves's user avatar
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1 answer
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Show that $S_1 \times S_1$ is not a cover of $S_2$

I am trying to show that the pair $(S_1 \times S_1, F)$ where \begin{align} F(\theta,\phi) = \{ \cos(\theta)\sin(\phi), \\ \sin(\theta) \sin(\phi), \\ \cos(\phi) \} \end{align} is not a cover of $...
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Monodromy for nonclosed paths

I've came across this proof of a theorem called "monodromy theorem" (in a completely non-analytical context) for closed paths that is as follows: Let $(Y,p)$ be a covering of $X$ and let $\...
ccnptr's user avatar
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$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point

Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
一団和気's user avatar
1 vote
2 answers
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Isomorphism between deck transformations and permutations onto the fiber

I came across this problem in Fulton's Algebraic topology textbook (problem 11.39) and I can't seem to get to the bottom of it. The problem is as follows: Let $p:Y\to X$ be a covering, with $Y$ ...
ccnptr's user avatar
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show that $P_* \pi_1(\tilde{X}, \tilde{x}_0)$ is exactly the given normal group

I was trying to understand this problem: Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. Draw a picture of the covering space of $S^1 \vee S^1$ ...
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Why we needed to prove that $f$ is the zero map before using the lifting criteria?

Here is the question I am trying to understand its solution and the hint given: Show that if a path-connected, locally path-connected space $X$ has $\pi_1(X)$ finite, then every map $X \to S^1$ is ...
Emptymind's user avatar
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Covering space being implied by finiteness of pre-image

I'm struggling to finish a proof about coverings. The question is as follow: Let $ p: E \to B $ be a local homeomorphism. Show that, if $ |p^{-1}(b)| = |p^{-1}(b')| < \infty $ for all $ b, b' \in B ...
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6 votes
2 answers
2k views

Why infinite spheres and not just two?

I was trying to understand this problem: Construct a simply-connected covering space of the space $X \subset \mathbb R^3$ that is the union of a sphere and a diameter. And my idea was to only use two ...
Emptymind's user avatar
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1 vote
0 answers
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A branched covering map is determined by the corresponding subgroup of the fundamental subgroup of the base

Let $f:X^n\to Y^n$ be a $d$-fold branched covering along branch locus $B\subset Y$. This is by definition a smooth proper map with critical set $B$ such that $f:X-f^{-1}(B)\to Y-B$ is a $d$-fold ...
blancket's user avatar
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1 vote
1 answer
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Proof of existence of universal cover

I was reading through a proof that a path-connected, locally path-connected, and semi-locally simply connected space $X$ has a universal covering space $\tilde{X}$ that I found on the internet here: ...
Confused's user avatar
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1 answer
28 views

Are covering spaces "classified" by a 2-dimensional CW complex?

Principal $G$-bundles are classified by maps into the classifying space $BG$: $$\mathsf{PFB}_B(G)\simeq [B,BG].$$ When $G$ is discrete, $BG=K(G,1)$ is an Eilenbeg-MacLane space, and a principal $G$-...
Alex Bogatskiy's user avatar
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What does one mean by a covering space $Y_1$ dominates another covering space $Y_2$

I was reading through the expository paper https://math.berkeley.edu/~dcorwin/files/etale.pdf In chapter 1 section 1.1.2 the statement says We have $H_1 ⊆ H_2$ iff $Y_1$ dominates $Y_2$ It basically ...
Anurenj ER's user avatar
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1 answer
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Whether covering transformation of universal covering have not fixed point?

Assuming $\pi: \tilde M\rightarrow M$ be a universal covering of a complete Riemannian manifold $M$. $f:\tilde M \rightarrow \tilde M$ is a covering transformation. If $f$ has fixed point, whether $f$ ...
Enhao Lan's user avatar
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1 vote
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27 views

Galois cover with reverse ramification data

Let $Y\to X$ be a Galois cover of complex varieties (or only consider complex algebraic curves) with the ramification data $(p_\bullet,\eta_\bullet)$, namely branch points $p_\bullet$ with a preferred ...
S.Gau at Math's user avatar

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