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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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Complex structure of branched cover over Riemann surface

Suppose $X$ is a Riemann surface, $Y$ is a Hausdorff topological space and $p: Y\to X$ is a local homeomorphism. Then there is a unique complex structure on $Y$ such that $p$ is holomorphic. Now if $\...
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Universal covering space of the real projective line?

I´m thinking about universal covering spaces. I´ve seen a lot of examples and authors ever say "the sphere $S^n$ is the universal covering space of the $n$-dimensional projective space $\mathbb{R}P^n$ ...
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Any compact orientable surface is a branched cover of a torus

Given a torus $T$, assume all the branch points are of index $2$, then by Riemann-Hurwitz theorem, the number of branch points is $2g-2$. Select $n:=2g-2$ points $\{x_i\}$ in $T$. Does that mean we ...
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Galois cover and solvability

This is a very general question: let $f:X\to \mathbb P^1$ be a branched cover of compact Riemann surfaces, where $X\subset \mathbb P^n\times \mathbb P^1$ and $f$ is the natural projection. For any ...
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If we add more relations to a presentation will it always form a quotient group?

Specifically, if I have a presentation $\left<G|R\right>$, and I look at the presentation $\left<G|R,R_1\right>$ is always true that $$\left<G|R,R_1\right>\cong\left<G|R\right>...
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Equivalence between Category of Covers and $\pi_1(X)$ Sets

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 38): In order to show the category equivalence claimed in Thm 2....
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When is exponential map from Lie algebra to Lie group a covering map?

Suppose $G$ is a Lie group and $\mathfrak{g}$ its Lie algebra. It is not so difficult to see that if $G$ is abelian and connected then $\exp:\mathfrak{g}\rightarrow G$ is a universal covering map. ...
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Hatcher problem 1.3.8

Suppose that $X$ and $Y$ are path connected and locally path connected spaces and that $\tilde{X}$ and $\tilde{Y}$ are simply connected coverings of $X$ and $Y$, repsectively. Prove that if $X$ and $Y$...
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Monodromy action

Let $p:E \to X$ a topological covering of connected space $X$. Fix a basepoint $x_0$ of $X$ and denote by $\pi(X,x_0) $ the fundamental group of $X$. The monodromy action of $\pi(X,x_0) $ on $p^{-1}...
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Smooth covering maps and the fundamental group

Let $M$ be a smooth, connected and locally path-connected manifold, and let $\pi: \tilde{M}\to M$ be its universal cover. Let $\text{Aut}_\pi(\tilde{M})$ be the group of smooth covering ...
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How to see correspondence between G-covers and homomorphisms

I found this paper on van Kampen's theorem (https://www3.nd.edu/~andyp/notes/SeifertVanKampen.pdf), and I was wondering how to prove Lemma 1: Let Z be a reasonable nonempty path-connected space, let ...
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Coverings of CW complexes are also CW complexes: How do I show that it has the weak topology?

Let $X$ be a CW complex, $p:E\to X$ a covering map. Then $E$ has an induced CW complex structure defined as follows. If $\Phi:D^n\to X$ is a covering, it lifts to a map $D^n\to E$ (since $D^n$ is ...
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Homotopy equivalence of covering spaces [duplicate]

I was trying to solve the following exercise in Hatcher (1.3.8). Let $p:(\tilde{X},\tilde{x})\to(X,x)$ and $q:(\tilde{Y},\tilde{y})\to(Y,y)$ simply-connected covering spaces. Assume $X,Y$ path-...
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Extending a covering space of the 2-skeleton of a CW complex to the CW complex

Let $Y$ be a CW-complex and $f: Y^2 \to Y$ be the inclusion of its two skeleton. We define $f^*Z := \{(y, z) \in Y^2 \times Z \ | \ y = f(y) = p(z)\}$ and $f^*p : f^*Z \to Y^2$ the restriction of the ...
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Identifying the branch points of a covering

Consider the compact Riemann surface $\overline{X}$, which is the compactification of $$X=\{(z,w)\in\mathbb{C}^2\mid z^{2a}-2w^bz^a+1=0\}.$$ Here, $a,b>0$ are integers. We let $p:\overline{X}\to\...
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Convering using path component [closed]

I have part (a) done. I'm having a hard time figuring out b and c. Any hints?
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Homotopy equivalence between quotients by free actions

Let $X,Y$ two contractible spaces. Assume there is a free action of a group $G$ on both spaces. $X$ and $Y$ are obviously homotopy equivalent. In particular, we can consider the homotopy equivalence ...
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How to construct a two-sheeted cover of a non-orientable surface?

Let $S$ be a non-orientable surface. Then there exists a two-sheeted covering map $p:S'\to S$ with $S'$ an orientable surface. I want to know how to construct $p$. I know that $\mathbb{R}P^2$ is 2-...
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Monodromy when fiber is connected

When I learned about monodromy, it was in the context of covering spaces. There, if $p$ is the covering map and $l$ is a loop with $\gamma(0)=\gamma(1)=x$, then a $\bar \gamma$ lift of $\gamma$ is ...
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Universal Cover of SO(n,C)

The corresponding question for the real special orthogonal group is well-known - these are the Spin groups. When one looks at complex special orthogonal groups though, this isn't the right way to go: ...
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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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1answer
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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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1answer
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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( ...