# Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

1,008 questions
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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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### Why do all the Platonic Solids exist?

In three dimensions it is quite easy to prove that there exist at most five Platonic Solids. Each has to have at least three polygons meeting at each vertex, and the angles of these polygons have to ...
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### Covering spaces - why are they useful?

As someone who trained as a physicist, I have known for ages that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so, during an idle day at the office I decided to look up what ...
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### Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose $X,Y$ ...
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### Why is the Long Line not a covering space for the Circle

I know of several reasons why the long line can't be a covering space for the circle, but I'm more curious in what exactly goes wrong with the following covering map. Let $L$ be the long line and ...
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### Universal cover via paths vs. ad hoc constructions

I'm looking for some intuition regarding universal covers of topological spaces. $\textbf{Setup:}$ For a topological space $X$ with sufficient adjectives we can construct a/the simply connected ...
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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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### Calculating monodromy

I'm right now learning about Monodromy from self-studying Rick Miranda's fantastic book "Algebraic Curves and Riemann surfaces". Today, I read about monodromy, and the monodromy representation of a ...
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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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### 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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### 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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### A good way to understand Galois covering?

A covering map $f:X\rightarrow Y$ is called Galois if for each $y\in Y$ and each pair of lifts $x, x^{'}$, there is a covering transformation taking $x$ to $x^{'}$. What is a good way to understand ...
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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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### 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 ...
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### Is a covering space of a manifold always a manifold

Assume $M$ is a manifold and $q : E \to M$ is a covering map. I have been told a few times that a covering space of a manifold is again a manifold. Indeed, it is easy to verify that $E$ is both ...
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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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### 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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### Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
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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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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 \... 1answer 267 views ### Is the universal cover of an integral homology sphere again an integral homology sphere? Let \Sigma be an integral homology sphere (from now on I will drop the word 'integral'). If \Sigma is simply connected, then it is homotopy equivalent to a sphere by Whitehead's Theorem, and ... 1answer 703 views ### The covering space of connected space Let X be a connected topological space, and \pi : Y \rightarrow X a surjective covering space map. Suppose that the group of deck transformations of \pi contains a subgroup \mathbb Z_p, where ... 2answers 88 views ### When a holomorphic function between hyperbolic surfaces is a covering map. I'm studying Milnor's book "Dynamics in one Complex Variable", and he states this problem to the reader in the middle of the proof of Pick's Theorem: If S and S' are two hyperbolic Riemann ... 1answer 2k views ### Lifting of maps to a covering space I am reading Algebraic topology by W. Massey and I have a problem with the proof of property 5.1: Let (\tilde{X},p) be a covering space of X, Y a connected and arcwise connected space, \tilde{... 1answer 272 views ### How to find fundamental groups and covering spaces of \mathbb{RP}^2\vee \mathbb{RP}^2? The following is an exercise I was assigned in homotopy theory. Defined X = \mathbb{RP}^2\vee \mathbb{RP}^2. a) Find \pi_1(X). b) Find the universal cover of X. c) Find all of its connected ... 0answers 2k views ### 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 \... 3answers 257 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? ... 3answers 786 views ### 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) ... 2answers 652 views ### Orientable double covers for non-orientable manifolds If I have two non-orientable connected manifolds such that their orientable double covers are homeomorphic, can anything be said about the manifolds? Are they homeomorphic? 4answers 4k views ### 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 ... 2answers 241 views ### The square map on SO(n) Let P:SO(n) \to SO(n) be the square map P(A)=A^2. Does P define a covering map structure? If yes, is there an action of a finite group G on SO(n) such that each fiber of ... 1answer 480 views ### Universal cover of T^2 \vee \mathbb{R}P^2  What is the universal cover of the wedge sum of the torus and the real projective plane? I know from Hatcher's Algebraic Topology that the universal cover of \mathbb{R}P^2 \vee \mathbb{R}P^2  is ... 1answer 1k views ### 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 ... 3answers 1k views ### 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 ... 2answers 582 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 \...
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 ...