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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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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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Bounding the cheeger constant of a cover of a graph?

Suppose that $G$ and $H$ are connected graphs, and that $\phi : G \to H$ is a $n$-fold covering. Let $h$ denote the Cheeger constant. I know that $h(G) \leq h(H)$. Question: Is it also the case that $...
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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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1answer
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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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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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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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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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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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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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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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$p:X\to Y$ 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
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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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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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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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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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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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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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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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1answer
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Involution and Covering space

Is there a connected topological space such that admits a free involution, trivial fundamental group and furthermore has the set of real number as it's covering space?
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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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2answers
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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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1answer
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Number of components in a covering space is less than or equal to number of sheets?

Context: on p235 of Hatcher's Algebraic Topology, Hatcher proves that if an n-manifold $M$ is connected, it is orientable if and only if its orientable double-cover $\tilde{M}$ has two components. ...
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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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Algorithm - Circle Overlapping

Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape ...
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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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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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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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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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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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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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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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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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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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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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1answer
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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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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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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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1answer
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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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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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1answer
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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 ...