# Tagged Questions

For questions about or involving covering spaces in algebraic topology.

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### Constructing covering space of surfaces

If $S_g$ is the surface $\#_g T^2$ where $g$ is a non-negative integer, when can we construct a covering space $S_h$ of $S_g$? Each such surface is a $CW$-complex, and in a $n$-sheeted covering, each ...
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### What are the morphisms in the category of unramified coverings over a compact Riemann surface?

Fix a compact Riemann surface $S$, and finite a set of branch points $B \subseteq S$. Consider the collection of Riemann surfaces $S_1$ and mermorphic functions $f: S_1 \rightarrow S$, such that $f$ ...
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### Algorithms for finding covering spaces of a given space

Taking the example of $X=S^1\vee S^1$ , to find the covering space $X$ what was done in Munkres is that we had the idea of how the real line wraps around the circle. Using this we attached circles ...
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### Covering space of a $\theta$ graph

I'm considering a problem of finding an explicit algorithm to construct a covering of a finite graph (in particular, of a $\theta$ graph) Since the current graph is homotopy equivalent to a wedge of ...
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### Geodesic Lines on Covering Maps

So I'm not sure how deck transformations work into this problem. I've established the following so far. Let $\pi:\tilde{M}\rightarrow M$ be the universal covering map. We may suppose that $M$ is ...
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### Is there is a way to construct a covering space of a wedge of two circles for a given normal subgroup

I would like to construct a covering space of a wedge of two circles with a given normal subgroup $H \subset \pi_{1}(S^{1} \vee S^{1})=F_{a,b}$. The goal is to find a covering space $\tilde{X}$ so ...
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### If p is a covering map of a connected space, does p evenly cover the whole space?

Suppose I have a covering map $p:X\rightarrow Y$, and $Y$ is connected. Is $Y$, as an open set, evenly covered by $p$? I think the answer is yes; I'm new to this kind of topology, so I'm not sure if ...
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### Is There a Generalization of the Path Lifting Property Of Covering Maps.

$\newcommand{\R}{\mathbf R}$ Let $p:(E, e)\to (X, x)$ be a covering projection map. We know that for any path $\gamma:I\to X$ such that $\gamma(0)=x$, there is a unique lift $\Gamma:I\to E$ such that ...
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### What happens if the Vector field vanishes in this case?

"Given the contravariant vector field $V{\mu}(x)$, we cosider te differential equation $$\frac{dx^{\mu}}{d\lambda}=V^{\mu}(x).$$ The solution $x^{\mu}$ is a map from $\mathbb{R}\rightarrow M$ is ...
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### Étale morphism has all its Deck transformation homotopic to identity

Is there an example that étale morphism (of degree $d,d<\infty$) $\pi: X\rightarrow Y$, s.t. all its Deck transformations homotopic to $Id_X$,except the trivial one, where $Y$ is general Enriques ...
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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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### Catalog of covering maps

Is there somewhere where I can find a list of covering maps including their base space and target space? Apart from standard examples found in notes and books I can't find much else.
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### Covering spaces of $S^1 \vee S^1$: to what subgroups do these ones correspond?

The universal covering space for $S^1 \vee S^1$ is the Cayley graph, $X$, of the free group on two generators, $F\{a,b\}$. The subgroup $F\{b\}$ corresponds to the covering space ...
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### Deck transformations of universal cover are isomorphic to the fundamental group - explicitly

I have read on several places that given a (say path connected) topological space $X$ and its universal covering $\tilde{X}\stackrel{p}\rightarrow X$, there is an isomorphism ...
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### Universal covering and double cover functors

Cross-posted on MO Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know ...
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### Why $f:\mathbb{C} \to \mathbb{C},~~~ z \mapsto z^3$ is not a covering map? [closed]

Can someone tell me why $f:\mathbb{C} \to \mathbb{C},~~~ z \mapsto z^3$ is not a covering map?
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### Covering Space Question

I recently encountered the following: Let $p:(E, e_0) \to (B, b_0)$ be a covering map. Assume that $p_∗(\pi_1(E, e_0)) \subseteq \pi_1(B, b_0)$ is a normal subgroup. If $e_1\in p^{−1}(\{b_0\})$, then ...
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### Definition of compactness unnecessarily verbose?

The definition of a compact set is given as a set, $X$, for which all open covers have a finite subcover. This seems unnecessarily verbose to me. Wouldn't it be sufficient to simply say that $X$ has ...
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### Make a complex polynomial a covering map

Let $p:\mathbb{C}\to \mathbb{C}$ be a complex polynomial. Let $C:=\{p(z):p'(z)=0\}$ and $V:=\mathbb{C}\setminus C$. I want to show that $p:p^{-1}(V)\to V$ is a covering map. By inverse function ...
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### Proving that the tangent vector of a simple closed curve rotates by $2 \pi$

I am trying to prove that if $\gamma(t)=(x(t),y(t))$ ,a function from the closed interval $[0,1]$ to $\mathbb{R^2}$ is a simple closed unit speed curve such that $\gamma '(0)=\gamma '(1)$. Then the ...
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### A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: ...
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### Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Let $P\to$ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?
Let $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ be covering maps. What would be an example that $q\circ p:X\rightarrow Z$ is not a covering map? I saw a counterexample here, but it was too complex. Is ...