# Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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### Smoothness of the action of the deck transformation group

In page 163 of John M Lee's "Introduction to Smooth Manifolds" (second edition), for a given smooth covering map $\pi:E\to M$, when the author proves the smoothness of the action of the deck ...
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### How to prove the continuity of the path lifting function of covering spaces.

Let $p\colon E \to B$ be a covering map. The path lifting function $\varphi\colon E \times_p B^I \to E^I$, where $E \times_p B^I := \{(e, \gamma) \in E \times B^I : \gamma(0) = p(e)\}$ is a pullback ...
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### Locally compact Hausdorff covering space

I am trying to prove the following conjecture: Let $\pi : E\to B$ be a covering map. If $E$ is locally compact Hausdorff, then so is $B$. (The converse is known to be true, cf. Exercise 6 in Section ...
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### Any smooth mapping from $\mathbb{R}^n$ into $S^1$ is of the form $e^{if(x)}$?

Let $F\colon \mathbb{R}^n \to S^1$ be a smooth mapping. Then, I strongly suspect that there must be a smooth function $f\colon \mathbb{R}^n \to \mathbb{R}$ such that F(x) = \exp \big( ...
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### Construct covering space with given fiber

Suppose $(X, x_0)$ has a universal cover and $A$ is a left $\pi$-set. How do I find a covering map $q: Y \to X$ with the fiber of $x_0$ isomorphic to $A$ as a $\pi$ set? I vaguely know that this has ...
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### covering space of torus [closed]

What is the covering space of the torus $T = S^1 \times S^1$ corresponding to the subgroup $2\mathbb {Z} \times 2\mathbb {Z}$ of $\mathbb {Z} \times \mathbb {Z}$? Here is my explanation, is it ...
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### Orientation covering

My question is about whether or not the image of an open set is still an open set. I'm going to write the construction of the orientation covering and then I'll ask what I can't figure out. Let $M$ be ...
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1 vote
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### Confusion about finding a covering space for $\langle a^2, (ab)^4, b^2\rangle \leq \pi_1 (\mathbb S ^ 1 \vee \mathbb S ^ 1)$ (Hatcher 1.3.12)

I am confused about the covering space of the wedge sum of two circles that corresponds to a subgroup $\langle a^2, (ab)^4, b^2\rangle$ (reminiscent of $D_4$) of the fundamental group of the base ...
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### Covering properties of non-constant holomorphic function $f: X \rightarrow \mathbb{C}$

I'm working through a proof that Riemann surfaces are second countable, and one of the main steps is showing that if $X$ is a connected Riemann surface such that there is a non-constant holomorphic ...
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### Fundamental group of covering space as the kernel of homomorphism

Consider a surjective homomorphism $\theta: Z_2 * Z_3 \to S_3$ ($S_3$ is the symmetric group on 3 objects) given by mapping the generators to elements of order 2 and 3 in $S_3$ respectively. By ...
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### Bijection involving the fundamental groupoid of a manifold

Let $M$ be a smooth manifold. I read in this post, that there is a bijection between the fundamental groupoid $\Pi(M)$ and $(\tilde{M}\times \tilde{M})/\pi_1(M)$, where $\tilde{M}$ is the universal ...
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### What does one mean by a covering space $Y_1$ dominates another covering space $Y_2$

I was reading through the expository paper https://math.berkeley.edu/~dcorwin/files/etale.pdf In chapter 1 section 1.1.2 the statement says We have $H_1 ⊆ H_2$ iff $Y_1$ dominates $Y_2$ It basically ...
Assuming $\pi: \tilde M\rightarrow M$ be a universal covering of a complete Riemannian manifold $M$. $f:\tilde M \rightarrow \tilde M$ is a covering transformation. If $f$ has fixed point, whether $f$ ...
Let $Y\to X$ be a Galois cover of complex varieties (or only consider complex algebraic curves) with the ramification data $(p_\bullet,\eta_\bullet)$, namely branch points $p_\bullet$ with a preferred ...