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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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752 views

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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How to prove that the center of the fundamental group of $T_g$ is trivial for $g \geq 2$?

Where $T_g$ is a closed orientable surface of genus g. I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization ...
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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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Given a curve over $\mathbb{P}^1$ how can I determine the monodromy around a ramification point?

Given a smooth curve of the form $$ f(y) - g(x) $$ over $\mathbb{A}^1_x$ with a smooth compactification $C \to \mathbb{P}^1$, how can I determine the monodromy of the points of the fibers? Are there ...
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When is the induced map $f^{*}:\operatorname{Spec}(S) \to \operatorname{Spec}(R)$ a covering map?

Let $f: R \to S$ be a ring homomorphism. When is the induced map $f^{*}: \operatorname{Spec}(S) \to \operatorname{Spec}(R)$ on topological spaces a covering map? More precisely, are there any ...
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How can I find the monodromy of a cyclic galois cover of the affine line minus a few points?

Consider the following cyclic covering of the affine line minus a few points: $$ \text{Spec}(\mathbb{C}[t,x]/(x^n - t(t-1)(t-2))) \to \mathbb{A}^1_t - \{ 0, 1, 2 \} $$ How can I find the local ...
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Where can I find precise examples of ramified coverings of $\mathbb{C}P^1$?

One of the definitions of simple Hurwitz number $h_{g,\mu}$ is that it counts up to automorphisms the number of ramified coverings of $\mathbb{C}P^1$ such that covering space is a connected surface of ...
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Is this covering group $G'$ of $G$ unique?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
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247 views

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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Finite-sheeted covers of integer homology spheres

For an integer homology sphere $M$, one can show that non-trivial connected finite-sheeted covers of $M$ must have at least $5$ sheets. This is because, for a given $n$-sheeted cover $M'$, one can ...
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Are there infinitely many non-negative integers not covered by one of these 7 polynomials?

Consider the following polynomials: $$ \begin{align} f_1(n, m) &= 30nm + 23n + 7m + 5 \\ f_2(n, m) &= 30nm + 17n + 13m + 7 \\ f_3(n, m) &= 30nm + 23n + 11m + 8 \\ f_4(n, m) &= ...
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152 views

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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Does a Deck Transformation Truly Shuffle?

Given a covering map $p:E \rightarrow B$, suppose $U \subseteq B$ is such that $p^{-1}(U) = \bigsqcup_{j \in J} V_j$ where $p$ maps $V_j$ homeomorphically to $U$. Then we want to say that whenever $f$ ...
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Construction of universal covering as an inverse limit as in Thurston's book

In Chapter 13 of Thurston's book on manifolds, he alludes to a particular construction of the universal covering of a space $X$, with the property that for all $x \in X$, there is a neighborhood $U$ ...
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How exactly does one construct a covering space corresponding to a subgroup

I am working throught Hatcher's Algebraic Topology and there are a number of exercises asking one to construct a covering space corresponding to a subgroup of the fundamental group, such as the ...
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Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
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Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
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Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
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Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over $T\...
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Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper (...
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Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
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How to construct certain cover given in Mumford's Abelian Varities book

In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
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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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Covering Space Examples

In my algebraic topology course we briefly discussed automorphism groups of covering spaces before moving onto deformation retracts. In passing, my professor mentioned that it is easy to come up with ...
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How to prove that $\Gamma$ is finitely generated?

Let $\pi \colon \tilde{M} \to M$ be the abelian cover, where $M$ is a compact, oriented smooth and finite-dimensional manifold. Let $\alpha \in \Omega^1(M)$ be a closed 1-form such that the kernel of ...
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52 views

3-fold simple branched cover

Where can I find a proof of the fact that every surface (with one boundary component) is 3-fold simple branched cover of the 2-sphere (2-disc)? I have seen this result in different places but without ...
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333 views

what's the universal covering group of general linear group $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$

Because the general linear group is not simply connected, they must be able to be covered by some simply connected Lie group. But I cannot imagine which Lie group with same dimension can cover $GL(n,\...
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Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
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Algorithms for covering a rectilinear polygon using rectangles of the same size

The following is the problem description: All angles of the polygon are right. It may be convex or concave. Use rectangles of the same size to cover the polygon. The edge of the polygon and ...
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58 views

Using Averaging and Deck Transformations to Compute de Rham Cohomologies of the $n$-Torus

I'm preparing for some comprehensive exams and this is a question from a past year. The $n$-torus $T^n = \mathbb{R}^n/\mathbb{Z}^n$ is both a smooth n-manifold and an Abelian group, by virtue of ...
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Range of Lifted Differential Operators

Let $D:C^\infty M \rightarrow C^\infty M$ a differential operator on a Riemannian manifold $M$. Define its maximal extension as closed operator $$D_\max:L^2M\supset \mathrm{dom}(D_\max):=\{u\in L^2M\...
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223 views

Quotient of universal cover by fundamental group

The circle $S^1$ has fundamental group $\pi_1(S^1)=\mathbb{Z}$, universal cover $\mathbb{R}$, and satisfies $S^1=\mathbb{R}/\pi_1(S^1)$. Similarly, $\mathbb{R}P^2$ has fundamental group $\pi_1(\...
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Is there a standard name for functions whose fibers are finite on every element in their image?

What would you call a function $f$ with the property that $\forall y\in \text{img}(f)\left(\left|f^{-1}[\{y\}]\right|\in \mathbb{N}\right)$? What about a function whose fibers all have the same ...
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What can be said about coverings when there is no universal cover?

I'm studying problems of classification and existence of coverings. There is a property that states that a universal cover is a covering of any other cover of the space. Also, in some sense one can ...
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Universal covering group of a semidirect product of Lie groups

I have a new relevant question. Let U, V, W be connected Lie groups, so that U is the semidirect product of V and W. By Schreier's theorem (Pontrjagin, Theorem 61), each group has a unique (up to an ...
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How are multivalued mappings composed?

As I understand it, there's basically two approaches to multivalued functions. One of them is that a multivalued function $\mathbb{C} \rightarrow \mathbb{C}$ is just a relation $\mathbb{C} \rightarrow ...
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339 views

Covering map is fibration

Let $f:X \rightarrow Y $ be covering map.Prove that $f$ is fibration. Proof: Let $Z$ be topological space, $g: Z \rightarrow X$ map and $F:Z \times I \rightarrow Y$ homotopy such that $F(z,0)=f(g(z))...
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If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
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Do all paths have a neighborhood about them which lift homeomorphically to a covering space?

Let $\widetilde{X}$ be a covering space of $X$ with projection map $p:\widetilde{X}\to X$. We know by definition that, for all $x$ in $X$, there exists a neighborhood $x\in U\subset X$ such that $p^{-...
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290 views

Euler characteristic of 2-sheeted covering space

I'm currently taking a course on algebraic topology and while doing exercises, I realised that I wanted to use the following: If $X$ is a compact connected $2$-manifold and $\varpi:Y \rightarrow X $...
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How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
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174 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
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Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
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does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
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886 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
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Connected Components of Covering Space

Let $\pi : \tilde M\to M$ be a covering map, $K_1, K_2\subset\tilde M$ are two different connected components and there exists such points $x\in K_1, y\in K_2$ that $\pi(x)=\pi(y)$. In other words, $\...
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129 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
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127 views

Characterization of maps with $\mathbb{Z}/2\mathbb{Z}$-equivariant lifts.

I'm interested in characterizing maps $f:\mathbb{R}P^k\to\mathbb{R}P^\infty$ that lift to a $\mathbb{Z}/2\mathbb{Z}$-equivariant map $\tilde{f}:S^k\to S^\infty$. For $k\geq 2$ I have been able to ...
3
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79 views

How to find every 4-sheet covering of the wedge sum?

Based on this question $4$-sheet covering of the wedge sum of two circles I know how to find one 4-sheet covering of the wedge sum, but how to find every 4-sheet covering of the wedge sum? I really ...