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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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Computing global monodromy of branched covering of Riemann sphere

Given an irreducible polynomial $F(X,Y)\in\mathbb{C}[X,Y]$, there is an associated Riemann surface $S$ obtained from the zero set of $F$. Projection onto the second coordinate gives a map $\pi:S\to \...
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Finitely sheeted covering space of a compact space is is compact

Let $p:X\to Y$ be a finitely sheeted covering space. I want to show that $X$ is compact if $Y$ is. I have proven the following lemma. Let $U\subset X$ be open containing $p^{-1}(b)$, then there ...
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First Cohomology of Abelian Cover

Let $S$ be a closed oriented surface. Consider the (universal) abelian cover $p \colon S_{ab} \rightarrow S$, i.e. the one whose group of deck transformations is the abelianization of $\pi_1(S,\star)$....
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Deck Transformations Question

Using the Hatcher's notation on proposition 1.39. Let $G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$ be the Deck transformation group for the covering space $p: \tilde{...
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Lift of an Open Neighborhood

On page 63 of Hatcher's book on Algebraic Topology, he says the following: ...one says $X$ is semilocally simply-connected if this holds. To see the necessity of this condition, suppose $p : \...
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Proposition 1.33 in Hatcher's Book

Recall the following proposition in Hatcher's Algebraic Topology: Proposition 1.31 The map $p_* : \pi_1(\widetilde{X},\widetilde{x}_0) \to \pi_1(X,x_0)$ induced by a covering space $p : (\widetilde{...
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Proper actions in complex geometry

[skip to next blockquote for the actual question] Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $L \subseteq \mathbb{C}$ on the ...
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Problem 1.3.12 from Hatcher

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 $S^1 \vee S^1$ corresponding to the normal subgroup generated ...
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Colorings of covers of graphs

Let $G$ be a finite graph. Let $Cov_k(G)$ be the set of all $k$-sheeted covers of $G$. Note that if $G$ is $\chi$-colorable, then so is every graph in $Cov_k(G)$. I am wondering how the average ...
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Fuchsian groups in $\text{SL}(2,\mathbb{R})$ and commensurability in $\text{GL}(2,\mathbb{R})$

Let $\Gamma_1,\Gamma_2 \subset \text{SL}(2,\mathbb{R})$ be two Fuchsian groups. Assume that they are commensurable as subgroups of $\text{GL}(2,\mathbb{R})$, that is, there exists $g \in \text{GL}2,\...
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Which smoothness properties are preserved under ramified covering maps?

Setting. Let $M$ be a Riemann surface and $\Gamma$ a discrete group that acts properly discontinuously on $M$ by holomorphic maps. It is well known that each $x \in M$ has a finite stabilizer, that ...
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possible number of sheets for a Moebius band covering

Let M be the Moebius band, identified by the quotient of $[0,1]\times [0,1]$ by the equivalence $(x,0) \sim (1-x,1)$. Let $p: M\to M$ be a covering and $n$ its number of sheets. Find the possible ...
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Monodromies of complex differential equation

Let $A:\mathbb{C}^*\to\mathrm{M}_n(\mathbb{C})$ be a holomorphic map. Consider the system of first order differential equations \begin{equation} \begin{cases} \frac{dY}{dz} = A Y\\ Y(1)= I, \end{cases}...
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is every path-connected covering of the Moebius strip a Galois cover?

Let $p : E → M$ be a covering of $M$ the Moebius Sttrip such that $E$ is path connected. Is this a Galois covering? My intuition is there must be some non locally path connected coverings that are ...
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Local homeomorphism + curve lifting property, gives a covering map

The Theorem 4.19. is from Otto Forster's Lectures on Riemann Surfaces. Theorem 4.19. Suppose $X$ is a manifold, $Y$ is a Hausdorff space and $p:Y\rightarrow X$ is a local homeomorphism with the ...
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Provide a three-fold connected non-regular covering of $\mathbb{S}^1 \vee \mathbb{T}^2$ along with its projection

Let $\mathbb{S}^1 \vee \mathbb{T}^2$ be the cicrle and the torus glued together at a single common point (wedge sum of the circle and the torus). I am asked to provide a (connected) triple non-...
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(Covering space) Does Deck move whole sheets in whole sheets?

Let $p: E \to B$ be a covering space. Let be $b \in B$, $U \in I(b)$ an evenly covered neighborhood of $b$, $U_i$ the sheets over $U$ such that $p^{-1}(U) = \coprod{U_i}$. If $\phi \in Deck(p)$, is ...
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If $f_0=\omega_m$ and $f_1=\omega_n$, why is it not automatic that $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$?

In the second paragraph of Hatcher's proof, fourth sentence, it says The uniqueness part of (a) implies $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$. How does the uniqueness part ...
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Homotopy classes of $X$ and that of its universal covering space.

I am reading 1.3 Covering space of Hatcher's 'Algebraic Topology' and I cannot find out what I am missing. In the book, he says "The advantage of this is that, by the homotopy lifting property, ...
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Does the covering involution lift to the pullback cover?

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$ such that $p:X \rightarrow Y$ is an etale Galois double cover. For a smooth projective variety $Z$ with $f: Z \rightarrow Y$ the ...
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Let $X$ be a connected CW complex and $G$ a group such that every $\pi_1(X)\to G$ is trivial. Show that every $X\to K(G, 1)$ is nullhomotopic.

Question 2 in Chapter 1.B in Hatcher's Algebraic Topology: Let $X$ be a connected CW complex and $G$ a group such that every homomorphism $\pi_1(X)\to G$ is trivial. Show that every map $X\to K(G, ...
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Does a deck transformation have a homotopy that lifts to it?

I have a closed connected manifold $X$, consider the universal cover $p: \tilde X \rightarrow X$. If I recall correctly, any homotopy $F: X \times I \rightarrow X$ with $F(\cdot, 0) = id_X$ lifts to a ...
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Does maps between fundamental groups induces a continuous map between spaces?

Main question is this: Suppos $M$ is a manifold and $G$ is a finite group. If there is a group homomorphism $\phi:\pi_1 M\to G$, is there a continuous map $f:M\to BG$, where $BG$ is a classifying ...
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Properties of covering spaces replacing base points by contractible subspaces

Let $(X,A)$ and $(Y,B)$ pairs satisfying the homotopy extension property (or a CW-pairs if necessary) with $A$ and $B$ contractible. Let $f:(X,A)\to (Y,B)$ a map of pairs and let $p_X:\widetilde{X}\to ...
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The fundamental group of the Lattice - (R x Z) U (Z x R)

I am trying to show that the identity map $ id:S_h \vee S_v \rightarrow S_h \vee S_v$ does not lift to L = $(\mathbb{R} \otimes \mathbb{Z}) \cup (\mathbb{Z} \otimes \mathbb{R}) $ via the covering ...
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This covering map is homeomorphism

Suppose that $f:\mathbf RP^2\rightarrow X$ is a covering map and $X$ is a CW-complex. Show that $f$ is homoemorphism. We know covering map is continuous and onto,so we should show that $f$ is one-...
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Are there more intuitive and/or illustrative reason for why covering maps do not compose besides the classical counterexample?

In an introductory topology class, one learns about covering maps $p : E \to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no....
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$\mathrm{LMod}(X)$ has finite index in $\mathrm{Mod}(X)$

I am reading about the Birman-Hilden conjecture, specifically the thesis of Rebbeca Winarski and there is some basic point I am not getting. We consider a covering space $p: Y \rightarrow X$ between ...
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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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 ...
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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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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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$p:X\to Y$ 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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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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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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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 ...