Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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41 views

Proof of relation between Betti numbers of base manifold and its covering space

For a covering $p\colon\widetilde{M}\to M$ with compact $\widetilde{M}$, how to show that $b_i(M)\leq b_i(\widetilde{M})$ for $0<i<n=\dim(M)$. If I understand this inequality correctly it says ...
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$SO(3)$ double covers $L(4,1)$

Let $P^2$ be the real projective plane. I am trying to show that its unit tangent bundle (for a fixed arbitrary metric on $P^2$) is a lens space $L(4,1)$. It seems that this paper (https://www.maths....
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Holomorphic n-th covering map between annulus must be $z^n$, essentially.

Let $A_R$ denotes the annulus $\{ z\in \mathbb{C}:1<|z|<R \}$. I guess and tend to prove the following(may not true): If $f:A_r\mapsto A_R$ is a holomorphic covering map with degree $n$, then $R=...
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56 views

Betti numbers of the orientable double cover is same as non-orientable one

I've struggled a little with the following claim quoted from (first line of) this post: The Betti numbers of $M$ is not changed after taking the orientable cover. Is this a valid fact? Any proof? ...
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Covering Transformations are covering maps theorem

If $(\widetilde{X_1},p_1)$ and $(\widetilde{X_2},p_2)$ be two covering spaces for $X$ then a Covering Transformation is a continuous map $F : \widetilde{X_1} \to \widetilde{X_2}$ such that $p_1 = p_2 \...
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Can the Klein bottle cover the torus? Show the Klein bottle can't be covered by a space $X$ with $\pi_1(X)=\mathbb{Z}/3\mathbb{Z}$.

I'm brushing up on covering spaces for an upcoming exam, and I came across the following problems related to the Klein bottle: Can there exist a cover of the torus by the Klein bottle? and Let $X$ ...
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1answer
62 views

Calculating $\pi_2(X\cup_\alpha e_\alpha)$ using Hurewics theorem and covering spaces

Consider the CW-complex $X$ obtained by wedging two circles. Denote by $a$ and $b$ the generators of $\pi_1(X)$. On $X$, attach two discs with attaching maps \begin{align*}S^1\stackrel{a^5(ab)^{-2}}{\...
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What is a g-frame?

I am reading A quick trip through knot theory (link to pdf!) by R.H. Fox, in particular the section of branched covering (section 8 pag. 26 into the document). When he describes his algorithm to find ...
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Construct a normal covering space.

Let $X$ be a connected and locally connected space. Let $(C,q)$ be a connected covering space over $X$. Then, can we construct a normal covering space $(E,p)$ over $X$ such that there exists a ...
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1answer
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Covering space of an interval is trivializable.

We say a covering space $(E,p)\to X$ is trivializable, if there is a space $F$ (with the discrete topology), and a homeomorphism between $\varphi:E\to X\times F$ such that $p=pr_1\circ \varphi$, where ...
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What are all the topological spaces that the $S^1$ can cover up to homeomorphism?

I'm currently studying $G$-coverings for the first time and I came across an interesting question. First of all, I know that all connected coverings of $S^1:=\{z\in\mathbb{C}\mid |z|=1\}$ are of the ...
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Question regarding isomorphism between quotient of fundamental groups.

I'm trying to solve the following problem that can be found in Kosniowski's a first course in algebraic topology. This problem is under the chapter on Borsuk Ulam theorem problem $20.7$ d. Question ...
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Universal covering

Let $M=S^1 \times \mathbb{R}^n$ with the metric $g=-d\psi^2+q$, where $d \psi^2$ is the standard metric on $S^1$ and $q$ is the euclidean metric on $\mathbb{R}^n$. I know that the universal covering ...
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1answer
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Can isomorphic groups act on a topological space in different ways?

The question arises from the fact that each topological manifold $X$ is homeomorphic to its universal cover $X_0$ quotiented by the action of the fundamental group $\pi_1(X)$. It is natural to ask ...
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Fundamental group of the complement of some quadric cones

Problem Consider the domain $$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$ and the map $$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+z_3^2]\;.$$ Define $U=\...
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62 views

Computing the group of covering translations of a covering space

I'm trying to understand covering spaces and covering projections. I'm looking at the following question: Consider the space $$Y = \{ (y,z) \in (\mathbb{C}\backslash \ \{0\}) \times S^1 | \frac{y}{|y|...
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Covering map associated to a fibration

I am reading about the generalization of the monodromy action to fibrations. In this settings, the lift of a path is not unique, but it is unique up to free homotopy. In particular there is a well ...
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1answer
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Finite sheeted connected covers of $T^2 \setminus D^2$

I have to find the surfaces (up to isomorphism) that are connected 3-fold covers of $X=T^2 \setminus D^2$. Using the Euler characteristic criterion, I can see that the Euler characteristic of a ...
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A Certain Degree 12 Covering of the Figure-8 Complement

Let $M=S^3\setminus K$ be the figure-8 knot complement. The fundamental group of $M$ has a presentation $\pi_1 M=\langle x,y\mid x^{-1}yxy^{-1}x^{-1}y^{-1}xyx^{-1}y^{-1}=1\rangle$. There is a ...
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About dimension of sheets of universal covering.

I need some clarification about the concept of $n-$sheet of a covering. A little disclaimer before anything else: ll the definition and images I'll use in this post comes from the Hatcher book. So a ...
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What are all three 2-fold coverings of the Klein bottle?

Let $K$ be a Klein bottle. I know that $$\pi_1(K)\cong \langle a,b | abab^{-1}\rangle.$$ All 2-fold coverings of $K$ will correspond to index $2$ subgroups of $\pi_1(K)$. Thus up to isomorphism, 2-...
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Help with moduli spaces of four-marked spheres

I'm thinking about the moduli space of the four-punctured sphere where some of the removed points are distinguishable and some are indistinguishable. I believe there should be some covering maps ...
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Lifting an involution to the universal cover

Let $X$ be a connected and locally simply connected topological space, and let $\widetilde{X}$ denote its universal cover. Assume additionally that the fundamental group of $X$ is finite, so that the ...
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A quotient space of a manifold by a covering space action is Hausdorff [duplicate]

Let $M$ be a topological manifold and suppose there is a covering space action of a group $G$ on $M$. This means that each $x\in M$ has a neighbrohood $U$ such that $g_1(U)\cap g_2(U)$ nonempty ...
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Covering space of a CW is also a CW , case of finite sheets

I have seen some answers online regarding this but I wanted to make sure that my proof was correct. I believe this only works if we assume that we have a finite sheeted covering Now we wish to see why ...
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2answers
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Definition of double covers

It's unusual that I have to ask for a definition on this site, but it in the context of spin structures, it seems common to talk about double covers without giving the definition. I understand the ...
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Why is this cover incorrect?

I have been trying to construct a 3-fold cover of the punctured torus. Specifically, I am trying to construct a once-punctured genus 2 surface. Here is an attempt. I know it's incorrect because I have ...
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A sheaf is locally constant iff the etale space is a covering

Let $X$ be a locally connected space a sheaf $F$ on $X$ is called locally constant whenever each $x\in X$ has a basis of open neighbourhoods$N_x$ and if $U,V \in N_x$ such that $U$ is a subset of $V$ ...
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1answer
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Generalization of Homotopy Lifting Property

To paraphrase the normal homotopy lifting property, it states that given a covering map $\pi:E\to X$, an interval $I=[0,1],$ a homotopy $f:Y\times I\to X,$ and a lift $\widetilde{f}_0:Y\to E$ of $f_0:...
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Different number of sheets of the cover implies there's not covering isomorphism

I have to prove that if $p_1:X_1\rightarrow Y$, $p_2:X_2\rightarrow Y$ are two covering maps (and therefore $X_1$, $X_2$ two covering spaces of $Y$), and the number of sheets is different, then $X_1$ ...
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Construct the covering homomorphism and the covering groups of $S^1$ having the form $R/pZ$.

Construct the covering homomorphism and the covering groups of $S^1$ having the form $R/pZ$ where $pZ$ is the set of integral mutliples of the integer $p$. This is the first part of an excercise from ...
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Construction of a Covering Space by 'Twisting'

I have a question about the explanantion of the idea behind the classiyfing spaces $BG$ with respect (topological) group $G$. In wikipedia is stated that The classifying property required of $BG$ in ...
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1answer
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Example of Finite Subcovers for Explicit Open Covers of $[0,1]$

Say we have the following open cover of $(0,1)\subset\mathbb{R}$: $$O=\bigcup_{n=2}^\infty \left(\frac{1}{n},1\right)$$ So this an open cover of $(0,1)$, but could I not turn it into an open cover of $...
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1answer
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Does Universal Property of Simply Connected Covering Spaces Require Local Path Connectedness?

I am trying to find out if the property that simply connected covering spaces cover other connected covers (the universal property) depends on local path connectedness. I know that the universal ...
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How to construct an open interval for $V_n'$?

Taken from the Munkres book page no :$ 333$ Theorem $53.1$: The map $p : \mathbb{R} \to S^1$ given by equation $ p(x)= ( \cos 2 \pi x , \sin 2 \pi x) $is covering map In proof munkres say that $...
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$P:E\to B$ & $f$ loop If there is a lifting $\hat{f}:I\rightarrow E$ of $f$ with $\widehat{f(0)}\neq\widehat{f(1)}$ then $f$ is not null-homotopic

Let $P:E\rightarrow B$ be a covering and $f:I\rightarrow B$ be a loop. If there exists a lifting $\hat{f}:I\rightarrow E$ of $f$ with $\widehat{f(0)} \neq \widehat{f(1)}$ then $f$ is not null-...
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1answer
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Let $(0,3)\rightarrow S^1$ be a map such that $x\in (0,3)\rightarrow e^{2\pi i x}$.

Let $(0,3)\rightarrow S^1$ be a map such that $x\in (0,3)\rightarrow e^{2\pi i x}$. Is this a covering?
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Characterizing a 3-manifold from branched covering over $S^3$ and viceversa

There is a theorem by Alexander that says that any closed oriented 3-manifold can be constructed as a branched covering of $S^3$ over a knot. But given a 3-manifold how do I find a branched covering ...
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Show that $\Phi$ is a covering map

Let $G$ be a locally connected topological group, $H < G$ a closed locally connected subgroup and $H_0$ the $H$ identity connected component. How can i show that $\Phi : G/H_0 \to G/H$ is a ...
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Is there a connected $n$-sheeted covering space of a closed orientable surface for any positive integer $n$?

Let $S$ be a closed orientable surface. For any positive integer $n$, is there a connected $n$-sheeted covering space of $S$? This is certainly not true if $S=S^2$ because $S^2$ is simply-connected. ...
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About a 5-fold cyclic branched cover of $\Bbb CP^1$ along 3 points

Suppose $F$ is a smooth connected complex curve, and we have a cyclic 5-fold branched cover $\pi:F\to \Bbb CP^1$ along 3 points. Let $\gamma$ denote the corresponding $\Bbb Z_5$-action on $F$. We have ...
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Covering map of the Cylinder

Question: Is there a covering map $\mathbb{R}^2 \to S^1 \times \mathbb{R}$ ? I picture the cylinder $S^1 \times \mathbb{R}$ as a roll of gift wrap that one "rolls the plane" onto (just a ...
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Calculating $H_1$ in branched covering

I am studying A quick trip to knot theory by Fox. In section 8 the author studies branched covering. I can follow the discussion but I cannot understand how he calculates the matrix $J^{\theta\rho}$. ...
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1answer
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The 3-component hopf link covering the trefoil

Let $H$ be the complement of the 3-component Hopf link, which is homeomorphic to the complement of the 3-chain link if you prefer, and let $T$ be the trefoil knot complement. I recently encountered ...
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1answer
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When do the local homeomorphisms given by a covering map extend to deck transformations?

Let $p: Y \rightarrow X$ be a (surjective) covering map and let $G$ be the deck group of $p$. Let $U$ open in $X$ be such that the disjoint open sets $V, W \subseteq Y$ are both homeomorphic to $U$ ...
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Sheaf of sections of a covering space is locally constant

I wish to prove that for $\pi : Y \to X$ a covering space (spaces involved are Hausdorff, semilocally simply connected, path connected, and locally path connected, among other things), the sheaf $$ \...
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Find all possible 3 fold cover of torus minus an open disk

I'm studying for my upcoming geometry qualifying exam and got stuck with this problem: Find all topological spaces that are 3 fold coverings of $S^1\times S^1$ minus an open disk. Here is my attempt: ...
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1answer
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How to show the well-definedness of a lift of a continuous map via a covering map?

Given a topological space $X$ is path-connected and locally path-connected. Say $p:Z \rightarrow Y$ is a covering map such that $p(z_0)=y_0$. A map $f:X \rightarrow Y$ is continuous and $x_0 \in X$ be ...
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Differential of a smooth covering map is always invertible?

If $f: E \to M$ is a smooth covering map between manifolds, is it true that the differential (or the push-forward) $df_p : T_p E \to T_{f(p)}M$ is invertible for every $p\in E$? I think this should be ...
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Lift of a shortest loop is length minimizing

Let $(M,g)$ be a $2$-dimensional oriented complete Riemannian manifold without boundary. Let $f:\big(\Bbb S^1,1\big)\to (M,x_0)$ be a smooth loop such that $f$ has minimum length among all admissible ...

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