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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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If a covering map has a section, is it a $1$-fold cover?

If $q: E\rightarrow X$ is a covering map that has a section (i.e. $f: X\rightarrow E, q\circ f=Id_X$) does that imply that $E$ is a $1$-fold cover?
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When is a local homeomorphism a covering map?

if $X$ and $Y$ are Hausdorff spaces, $f:X \to Y$ is a local homeomorphism, $X$ is compact, and $Y$ is connected, is $f$ a covering map? It seems to be, and I almost have a proof, but I'm stuck at the ...
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Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$? [duplicate]

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
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1answer
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composition of certain covering maps

This problem was posted before, but not the proof (because the asker knowed the answer), only a counterexample without the hypothesis of finite fibres. I want to know how to prove this proposition: ...
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4answers
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Why this map is a covering map?

I'm trying to find the universal covering space of the Klein bottle. I know that $\mathbb R^2$ covers the Klein bottle , but I don't know how to prove, I found this proof on internet: Someone knows ...
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Is composition of covering maps covering map? [duplicate]

In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that ...
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A proper local diffeomorphism between manifolds is a covering map.

The following is an exercise taken from "Manifolds and Differenial Geometry" by Jeffrey M. Lee. Let $\widetilde M$ and M be (connected) $C^r$ manifolds. Let $f: \widetilde M \to M$ be a proper map ...
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Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
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1answer
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covering spaces and the fundamental groupoid

Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between ...
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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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2answers
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Covering space of a non-orientable surface

I have the following problem: Find the 2-sheeted (orientable) cover of the non-orientable surface of genus g. The cases $g=1,2$ are well-known, we have that the cover of $\mathbb{R}P^2$ is $\...
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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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Cayley complex as universal covering space

In Combinatorial Group Theory, Lyndon and Schupp construct a complex $K(X;R)$ from a presentation of group $G=(X;R)$, such that $G \simeq \pi_1(K,v)$ (proposition 2.3, p.117). Moreover, the Cayley ...
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The fiber of a covering space over a connected space has constant cardinality

Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for every $b \in B$. I know that $E$ has a ...
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1answer
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covering map with finite fibres and preimage of a compact set

Let $f:X\to Y$ be a covering map (covering maps are surjective) , Y be compact set. And suppose that $f^{-1}(y) $ is finite for each $y\in Y$. Prove that $X$ is also compact. I think that this ...
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The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent [duplicate]

This is exercise 1.3.8 in Hatcher: Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then $\...
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Classifying Covering Spaces using First Cohomology

I am familiar with the classification of covering spaces of a space $X$ in terms of subgroups of $\pi_1(X)$ (up to conjugation). However, if $X$ is a manifold, I know that $H^1(X; G)$ classifies G-...
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1answer
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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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3answers
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Is a path connected covering space of a path connected space always surjective?

If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where ...
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A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: If ...
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1answer
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Is there is a way to construct a covering space of a wedge of two circles for a given normal subgroup

I would like to construct a covering space of a wedge of two circles with a given normal subgroup $H \subset \pi_{1}(S^{1} \vee S^{1})=F_{a,b}$. The goal is to find a covering space $\tilde{X}$ so ...
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1answer
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Is a covering space of a completely regular space also completely regular

I'm trying to solve a problem in Munkres' Topology book. Let $p: E \rightarrow B$ be a covering map and suppose that $B$ is completely regular (for any closed subset $A$ and disjoint point $a$ there ...
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1answer
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Showing the Sum of $n-1$ Tori is a Double Cover of the Sum of $n$ Copies of $\mathbb{RP}^2$

I want to show that the non-orientable surface of genus $n$ has a 2-sheeted cover by an orientable surface of genus $n-1$. The base cases are easy: $S^2$ covers $\mathbb{RP}^2$ and I worked on a ...
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1answer
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Is the quotient $X/G$ homeomorphic to $\tilde{X}/G'$?

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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2answers
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Universal covering spaces of homotopy equivalent spaces are homotopy equivalent

I'm working on an exercise from Hatcher's Algebraic Topology (exercise 1.3.8): Let $\tilde X$ and $\tilde Y$ be simply-connected covering spaces of the path-connected, locally path-connected spaces ...
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1answer
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Two covering spaces covering each other are equivalent?

Let $X_1,X_2,X$ be connected and locally path-connected spaces and $p_1:X_1\to X$ and $p_2:X_2\to X$ be covering maps. Suppose that there are maps $f:X_1\to X_2$ and $g:X_2\to X_1$ such that $p_1=...
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Uniqueness for a covering map lift: is locally connected necessary?

So I just got through proving the following theorem: If $p:C\to X$ is a covering map and $Y$ is a [xxx] space, then given $y_0\in Y$, $c_0\in C$, $f:Y\to X$ such that $f(y_0)=p(c_0)$ there exists ...
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1answer
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Induced maps in homology are injective

In a topological course I have learned that a covering map $p:\bar{X}\rightarrow X$ induces a map on fundamental groups and that this induced map is a monomorphism. Now I wonder myself if this is ...
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1answer
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this map $p(z)=z^n$ is a covering map?

For any positive integer $n$ the map $p:S^1\to S^1$, defined by $p(z)=z^n$ is a covering map. I think it's easy, but as I'm a really beginner I'm struggling to prove it. By the way, what kind of ...
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1answer
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The universal cover of a path-connected, locally path-connected space $X$ covers any other covering space

I'm currently reading Hatcher's Algebraic topology book. In page 68 he says: A consequence of the lifting criterion is that a simply-connected covering space of a path-connected, locally path-...
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1answer
389 views

Can two different topological spaces cover each other?

I.e. do there exist non-homeomorphic $X$, $Y$ and covering maps $f:X\rightarrow Y$, $g: Y\rightarrow X$? I have a basic understanding of covering space theory as its taught in school. I was inspired ...
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Prove every map from the projective plane to the circle is nullhomotopic

Prove that every continuous map $f:P^2\to S^1$, where $P^2$ is the projective plane, is nullhomotopic. I think I need to use the fact that $\pi_1(P^2) = \mathbb{Z}/2\mathbb{Z}$ and covering space ...
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1answer
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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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1answer
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How to classify 3-sheeted covering space for $S_{1}\vee S_{1}$?

This might be a duplicate. This question also feels routine (it is also the execrise 10, page 88 in Hatcher). From Harvard qualification exam, 1990. Let $X$ be figure eight. 1) How many 3-sheeted, ...
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1answer
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Monodromy correspondence

Lately I've been studying monodromy and covering maps (in particular ramified covering mapos of Riemann surfaces), and I came across something I didn't fully understand. Let $V$ be a connected real ...
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1answer
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Prove that a covering map is a homeomorphism

I got stuck in the following exercise: Let $p:\widetilde{X}\rightarrow X$ be a covering map with $\widetilde{X}$ connected and $p^{-1}(x)$ finite, for every $x\in X$. Show that if there exists a ...
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4answers
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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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1answer
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Universal Cover of projective plane glued to Möbius strip

Consider the usual cell structure on $\mathbb R P^2$, with one 1-cell and one 2-cell attached via a map of degree 2. Consider the space $X$ obtained by gluing a Möbius band along the 1-cell via a ...
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proving that a covering map with certain domain and range is homeomorphism

Let $p:E\to B$ be a covring map, with $E$ path connected. Show that if B is simply connected, then $p$ is a homeomorphism. Well I don't know exactly what can I do here, maybe I have to start with ...
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Homotopy equivalent spaces have homotopy equivalent universal covers

A problem in section 1.3 of Hatcher's Algebraic Topology is Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. ...
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1answer
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Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
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If a connected open set is evenly covered, then its preimage is uniquely partitioned into slices

This is from Topology by Munkres: Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ ...
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1answer
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Compact subspace of a covering space

I've been working through Massey's A Basic Course in Algebraic Topology and I've gotten stuck on the following exercise (V.8.4): Let $X$ be a regular topological space, and $(\tilde{X}, p)$ a ...
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1answer
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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 ...
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1answer
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Map $f:S^2 \to S^1$ with $f(-x) = -f(x)$

Consider a continuous function $f:S^2 \to S^1$ with $f(-x) = -f(x)$ for all $x \in S^2$. I intend to show that such map doesn't exist using topological covering/lifting theory. Here my attempts: If ...
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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 help ...
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1answer
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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 ...
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2answers
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Is a polynomial also a covering map?

Question: Let $p(z)$ be a polynomial over $\mathbb{C}$. Is it true that $p:\mathbb{C} \to \mathbb{C}$ is a covering map ? Partial answer: Let us look first at the points where $p'(z)\ne 0$. There the ...
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1answer
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Universal Cover of a Surface (with Boundary)

I'm trying to see if there is a "nice-enough" way of describing/constructing the universal cover for a compact surface with $n$ boundary components. Clearly, if $n=0 $, the classification theorem for ...
4
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2answers
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Does $\pi_1(X, x_0)$ act on $\tilde{X}$?

Let $X$ be a path connected, locally path connected space and let $p:\tilde{X} \to X$ be a covering map. Let $x_0\in X$. Then we have a natural right action of $\pi_1(X, x_0)$ on the fibre $p^{-1}(x_0)...