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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

4
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1answer
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Show that a finite generated subgroup $H$ of $G$ with index 3 exists which is not normal in $G$.

> Suppose that $a$ & $b$ are the generators of a free group $G$.Show that a finite generated subgroup $H$ of $G$ with index 3 exists which is not normal in $G$. The way i try to solve the ...
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34 views

Covering spaces for $S^1 \vee S^1\vee S^1$

When I read the HATCHER book(algebraic topolog) a question i found on a pdf and the truth is i cant find a way to solve it and realize it! the question which i saw after reading page 57/58/59( ...
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19 views

Degree of universal cover of simple Lie group

I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie ...
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7 views

Lie group map whose differential is an isomorphism is a covering map

While trying to read the proof in Fulton and Harris of their “Second Principle,” I ran across something that I do not understand. They seem to claim that if $f: G\rightarrow H$ is a map of Lie groups ...
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31 views

Deck transformations acting on fundamental group

Let $Y$ be a space with $\pi_1(Y)=\mathbb{Z}^{2n}\times\mathbb{Z}^{2n}\times\mathbb{Z}^{2n}\rtimes_\psi S_3$, where $\psi: S_3\to Aut(\mathbb{Z}^{2n}\times\mathbb{Z}^{2n}\times\mathbb{Z}^{2n})$ by ...
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17 views

Irregular covering of 2-holed torus $S_2$

I need to find an irregular covering of the 2-holed torus. If a covering is regular, then $p_*(\pi_1(E,e_0))$ is a normal subgroup of $\pi_1(S_2,b_0)$, where $E$ denotes the covering space. So an ...
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2answers
37 views

Let $q\colon E\to X$ be a covering map. If $E$ is compact then $X$ is compact and $q$ is a finite-sheeted covering.

Here's what I have so far: Since $q$ is continuous and surjective then $X$ is compact. For all $x\in X$ there exists an evenly covered neighborhood $U_x$, so $\{U_x \colon x\in X\}$ is an open cover ...
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2answers
22 views

pull back of smooth covering space is injective

I need to prove this but I don't really know where to start: Let $p:M\to N$ be a smooth covering space between smooth manifolds. Show that $p^*:\Omega(N)\to\Omega(M)$ is injective. Where $\Omega(M)$ ...
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1answer
24 views

Cover of a simply connected and lpc space is trivial

I have I question about a reduction step in the proof of the statement that a cover of simply connected and locally path-connected space is trivial (source: "Fundamental Groups and Galois Groups" by ...
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0answers
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Covering with total space a CW complex

Let $G$ be a discrete group acting properly discontinuous on a CW complex $X$. Does $X/G$ also have a CW structure? Further questions: If not, maybe with stronger conditions ($G$ finite, each ...
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1answer
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regular covering transformation

can you help me with proving this statement? > Suppose that $F:S^1\longrightarrow S^1 $ such that $F(z)=z^n$ , is a covering space.prove that $F$ is regular covering transformation Honestly..i ...
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1answer
50 views

covering spaces and equivalency of these three propositions [closed]

This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following ...
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2answers
56 views

Classification of covering spaces for spaces that are not locally path connected: counterexamples?

The standard theory treats the case where the base space $B$ is path connected, locally path connected, and semi-locally simply connected. While being path connected and semi-locally simply connected ...
2
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1answer
32 views

Fundamental group via deck transformations considering rotation on sphere

Let $Z_m$ act on $S^1$ by multiplication with $e^{2\pi ki/m}$ for $k \in Z_m$. Let $X = S^1 / Z_m$ be the orbit space of this action. Then we have a universal cover $q:S^1 \rightarrow X$ given by the ...
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1answer
37 views

Uniqueness of lift in covering space theory

Let Y be a topological space and $\pi: X\rightarrow Y$ a covering map. Take two lifts of the covering map $\tilde {\pi_1}$, $\tilde {\pi_2} : X\rightarrow X$ such that they agree on $x_0 \in X$. ...
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60 views

Infinite cyclic cover corresponding to non-zero cohomology class $\alpha \in H^1(x,\mathbb Z)$

I want to understand the following sentence: Let X a compact (complex) manifold which has a non-zero cohomology class $\alpha \in H^1(X,\mathbb Z)$. Let $\pi: \bar X\to X$ be the corresponding ...
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1answer
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Deck transformation group in algebraic geometry

Let $f:X\to Y$ be a finite morphism between (irreducible) varieties. We can define $\operatorname{Aut}(X/Y)$ to be the automorphism of $X$ commutes with $f$. For the case over $\mathbb C$, we can also ...
2
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1answer
112 views

Covering map corresponding to surjection

Let $X$ be a compact manifold with $\pi_1(W)=S_3$, $T$ be an $n$-dimensional complex torus. Denote the universal cover of $W$ by $\tilde{W}$. Let $S_3$ act on $(T\times T\times T) \times \tilde{W}$ ...
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1answer
55 views

Index of covering in covering group

In wikipedia I read: In mathematics, a covering group of a topological group $H$ is a covering space $G$ of $H$ such that $G$ is a topological group and the covering map $p : G \rightarrow H$ is a ...
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1answer
58 views

$p:X\to Y$ covering space if $q\circ p:X\to Z$ and $q:Y\to Z$ covering spaces and $Z$ locally path-connected.

I have done the problem, but I'm confused about why the local path-connectedness of $Z$ is necessary. My solution: Let $p:X\to Y$ and $q:Y\to Z$ be such that $q$ and $q\circ p$ are covering spaces. ...
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1answer
30 views

What maps descend to homeomorphisms

I was reading "A primer on Mapping Class Groups" by B. Farb and D. Margalit and I'm stuck at a point in the proof of Mod$(A)\cong \mathbb{Z}$ where $A$ is the annulus, where the restriction of $M$ on $...
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2answers
100 views

Curves on surfaces lifting to embedded curves in finite covers

Let $S$ be a orientable closed surface with genus $g \geq 1$ and let $\gamma \subset S$ be an immersed curve. Does there exist a finite cover of $S$ where $\gamma$ lifts to a curve that is homotopic ...
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0answers
37 views

Fundamental group action on universal cover chain complex

I got a little bit confused with Hatcher's explanation of fundamental group action on the chain complex of the universal cover (Algebraic Topology, A. Hatcher, p. 328). Assume $X$ is a path-connected ...
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0answers
27 views

Give finite graphs or topology spaces $X_1,X_2$ which satisfying the following properties. [duplicate]

Try to give two finite graphs (or topology spaces) $X_1$ and $X_2$, such that: (a) There exist a graph (or topology spaces) $X'$, such that $X'$ is the finite-sheeted covering space of both $X_1$ and ...
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0answers
30 views

Covering space: preimage

Let $A = \{(x,y) \in \Bbb C^2: y^2 = f(x)\}$ where $f$ is a polynomial of degree $d$ without repeated roots. Let $f: A \to \Bbb C$ be defined by $f(x, y) = x$. For large $R$, what is the preimage ...
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1answer
27 views

Fibers Under a Covering Map are Discrete Subspaces of the Domain

In Munkres' topology book, the following claim is made: If $p : E \to B$ is a covering map, then for every $b \in B$, $p^{-1}(b)$ is a discrete subspace of $E$. Here's my attempt at a proof: ...
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0answers
31 views

Uniqueness of a time-orientable covering spacetime

On a standard treatment of time-orientability in Lorentzian spaces, it is usual to see a remark of the form: "If a space-time (M,g) is not time-orientable, then it has a double covering space which is....
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2answers
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An alternative, more formal proof of a path lifting criterion in tom Dieck's Algebraic Topology

This is a theorem from Tammo tom Dieck's Algebraic Topology: While it has a direct proof, the author gives a more formal proof in the problems: By pullback I suppose he means a diagram $\require{...
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0answers
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What is the relation between torsion elements of the class group and covering spaces of curves?

For a Dedekind domain $A$ we have the following result relating torsion elements of the class group to (mostly) unramified extensions: If $a\in A$ is such that there exists an ideal $I$ with $I^n=(a)$...
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35 views

The fundamental group of a topological space is isomorphic with its connected component fundamental group

can you help me with this problem of fundamental groups? suppose that $X$ is a topological space, let's fix a point on $X$ like $p\in{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ ...
0
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1answer
19 views

coverage mapping at covering spaces

the professor at university asked us: is a coverage mapping like P from X to Y a closed mapping or not. Also; is p an open mapping? i could prove that P is an open mapping but for proving that P is ...
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1answer
49 views

Why does the covering transformation group act properly discontinuously on the fiber?

In Step 2, I don't understand the part "for otherwise, two points would belong to the same orbit and the restriction of $\pi$ to $U_\alpha$ would not be injective". What two points? If $g\ne e$ and $...
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0answers
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Regular Covering of $S^1 \vee S^1$ with genus 1

The question is to prove that every finitely generated free group can be realized as a normal subgroup of the free group F(2). I want to do it by considering regular covers of $S^1\lor S^1$. For the ...
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1answer
74 views

Lemma 81.1 in Munkres

In the proof of Munkres, which is given in the picture below, I don't understand why we need to show that $h,\ h(e_0)=e_1$ exists iff $[\alpha]\in N(H_0)$. So I tried to organize the proof in the way ...
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1answer
34 views

The covering transformation is determined by its value at the base point

Consider the group of covering transformations of the (based) covering map $p:(E,e_0)\to (B,b_0)$. On p. 488 Munkres writes that a covering transformation $h$ is uniquely determined by its value at $...
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1answer
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If i have two closed and disjoint sets A and B in a set X c $R^{m}$, are there any open and disjoint sets in X that contain A and B?

I have a set $X$ $c$ $R^{m}$ and a pair of two closed and disjoint sets in X, called A and B. What i am trying to found at least a pair of sets Y and Z that are open and disjoint in a set X c $R^{m}$ ...
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2answers
176 views

Ramified Cover of Affine Scheme

I have some problems in geometrically visualization/ interpretation of the morphism $$f: {\displaystyle {\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])}$$ that is induced by canonical ...
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2answers
38 views

How to check if a cover is normal given the action on a fiber.

Assume you have a space $X$ and a homomorphism $r:\pi_1X\to S_n$. Then $r$ defines a n-sheeted covering space $Y$ of $X$. My question is, can you immediatly determine if $Y$ is a normal covering ...
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1answer
78 views

Classification of double coverings and $H^1(X ; \mathbb{Z}/2\mathbb{Z})$

It is a well known fact (see for example Hatcher's "Algebraic Topology", chapter $1$) that there is a bijection between the $n$-sheeted coverings of $X$ up to isomorphism of covering spaces and the ...
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1answer
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the fundamental group of $X$ is the symmetric group $S_3$, then whether it has a universal cover?

Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover. Try: Based on Hatcher, $X$ has a ...
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0answers
38 views

Trouble in finding the Galois group of the covering spaces of $S^1 \vee S^1$.

I am studying covering spaces and deck transformations from the book Algebraic Topology written by Allen Hatcher. While reading deck transformations I came across the concept of Galois group of ...
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1answer
29 views

Group of covering Transformation corresponding to proper discontinuous action of a group on a connected space is the group itself.

Suppose $G$ acts properly discontinuously on a connected space $X$. Show that the group $G(X,p,X/G)$ of the covering transformation of $p:X \rightarrow X/G$ is $G$. I have tried in this manner- ...
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1answer
85 views

Which surfaces does a compact orientable surface with boundary cover?

Suppose I have a bounded, orientable genus 5 surface with 4 boundary circles. Is there a way to determine what surfaces it covers? First, I know that there is a covering map from the closed ...
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19 views

Homotopy fibre of pushout

Let $(f_U,f_A,f_V):(U\leftarrow A\rightarrow V)\to (U'\leftarrow A'\rightarrow V')$ be a morphism of diagrams of topological spaces and let $f:U\sqcup_A V\to U'\sqcup_{A'} V'$ be the resulting ...
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1answer
60 views

Prove that quotient map is covering map

I'm self-studying algebraic topology and need help with the following problem (I'm only at part a.) The relevant definitions are as follows. Definition: Let F be a discrete space and X be any ...
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1answer
63 views

Are contractible open subsets always evenly covered?

Let $p:E\rightarrow X$ be a covering map. Let $U$ be a contractible open subset of $X$. Is $U$ necessarily an evenly covered open subset, i.e. $p^{-1}(U)$ a disjoint union of open subsets of $E$ ...
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2answers
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What's the definition of coherent topology in Munkres Topology?

Munkres Topology. In Section 71, coherent is "if" but on Wikipedia (Coherent topology), it's "if and only if" This can be seen in Example 1 of Section 83 We suppose $D \cap A_{\alpha}$ is closed in $...
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1answer
32 views

Understanding noncyclic covers of knot complements

Let $K$ be a knot in $S^3$. I'm familiar with the process of taking the $n$-fold cyclic cover $X_n(K)$ of the knot complement $X(K) = S^3 - K$ and I know that this cyclic cover can be completed to a ...
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0answers
46 views

Find all degree 3 covering spaces of the torus (up to isomorphism) [closed]

Find all degree 3 connected covers of the torus (up to isomorphism) I saw this as a part of part of a part of a question, so there should probably be a simple solution. Unfortunately, I don't see ...
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0answers
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Double cover of $\operatorname {SO}(V\oplus V^*) $

We know that $\operatorname{Spin}(V \oplus V^*)$ is the double cover of $\operatorname {SO}(V\oplus V^*)$ via the map $$\rho: \operatorname {Spin}(V \oplus V^*)\rightarrow \operatorname {SO}(V\oplus V^...