# Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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### Understanding the universal covering of certain surfaces

In a lecture of Algebraic Topology we were given a few examples of universal covering. One of them was the universal covering of connected surfaces with no boundary, saying that its universal covering ...
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### Fundamental group acts on universal covering by isometries

I am reading the proof of Theorem 4.13 in part II in the book Metric Spaces of Non-Positive Curvature of Martin R. Bridson and André Haefliger and in the proof they write "Consider the universal ...
1 vote
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### Pullback of a covering map along the covering map.

Consider $H$ as a subgroup of $G$, we can push them to the level of classifying spaces $p: BH \rightarrow BG$. This is a covering space with fiber $[G:H]$. What is the pullback of this covering map ...
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### A covering space of the mobius band

I was doing Hatcher exercise 1.3.21 recently and I failed to figure out the covering space of the space Y formed by attaching a mobius band to $\mathbb{RP^2}$. So I do some research on mathstackage ...
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### Space without universal cover (Solution Exercise A. Hatcher 1.3.5)

Problem: Let $X$ be the subspace of $R^2$ consisting of the four sides of the square $[0, 1]×[0, 1]$ together with the segments of the vertical lines x = 1/2, 1/3, 1/4, ··· inside the square. Show ...
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### The covering space of a completely regular space is also completely regular.

The Problem: Let $p: E\to B$ be a covering map. If $B$ is completely regular, so must $E$. The question has been asked before here and here. The main thing I am confused about is, for example, the ...
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1 vote
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### Characteristic Property of the Orientation Covering (Problem 15-10 in Lee's Smooth Manifolds book)

The problem statement: Let $M$ be a connected nonorientable smooth manifold with or without boundary, and let $\widehat{\pi}: \widehat{M} \to M$ be its orientation covering. Prove that if $X$ is any ...
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### Algebraic G -Galois Branched Coverings

Let $f: X \to Y$ a finite surjective morphism of integral normal projective varieties over some field such that the corresponding extension of function fields $K(Y) \subset K(X)$ is Galois with Galois ...
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### Lifting triangulation by branched covering map

Let $X, Y$ be compact connected manifolds of the same dimension, and let $f : X \to Y$ be a branched covering with finitely many branch points. (For example, $f$ could be a holomorphic function ...
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### Deck transformation Calculations.

Is there any source for the detailed calculations for the Deck Transformations stated on pg.70 in AT given below:
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### What's the deck transformation group of quotient space?

The question arises from the proof of Galois covering theorem, the one describing the one-one correspondence of intermediate coverings (equivalent class) and subgroups of the deck transformation group ...
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### Showing that $X$ has a unique topological group structure.

Here is the question I am trying to solve: Let $G$ be a topological group and $p: X \to G$ be a covering. Let $e \in G$ be the identity element of $G$ and choose $f \in p^{-1}(e).$ $(a)$ Show that $X$ ...
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### Successive quotienting of deck transformations

Let $\widetilde{X}\to X$ be a covering space, and let $G=D(\widetilde{X}\to X)$ denote the group of deck transformations. In this question is is shown that the space $\widetilde{X}/G$ is also a ...
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### When does a covering map between boundaries extend to covering maps between interiors?

Given two smooth three-manifolds $(M,\partial M)$ and $(N,\partial N)$ with smooth boundary, if we know that there is a covering map $\Gamma:\partial M\to\partial N$, is there a simple criterion to ...
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1 vote
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### Why are groups of type $F_2$ finitely presented?

There are some equivalent definitions for "finiteness properties", but let's define that $G$ is a group of type $F_n$ if it is the fundamental group of a CW complex $X$ whose $n$-skeleton is ...
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### Prove that each $R \in SO(3)$ has exactly 2 preimages with respect to the covering map $\Phi_U : SU(2) \rightarrow SO(3)$

Prove that each $R \in SO(3)$ has exactly 2 preimages with respect to the covering map $\Phi_U : SU(2) \rightarrow SO(3)$ I tried in this way: Let's consider $U \in SU(2)$ and $X \in V = i su(2)$ \...
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### Local Degree of smooth map vs Degree of a Covering

Consider any two oriented connected closed smooth manifolds $M, N$ of same dimension $n \ge 1$. Let $f : M \to N$ be a smooth map. Recall that a regular value of $f$ is a point $y \in Y$ such that the ...
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### Riemann surfaces and coverings

Suppose i have two riemann surfaces $S_{g_1}$ $S_{g_2}$ of genus $g_1$ and $g_2$, i want to find a criteria when first surface covers the second. My guess is that $2-2g_1 = k(2-2g_2)$(euler ...
1 vote
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### What is the induced homomorphism of the covering map $p : Y \to X : z \mapsto z^3 - 3z$ on the fundamental groups?

Let $X = \mathbb{C} \setminus \{ \pm 2 \}$ and $Y = \mathbb{C} \setminus \{ \pm 1, \pm 2 \}$. The map $$p : Y \to X : z \mapsto z^3 - 3z$$ is a 3-branched covering as given in this question of Math ...
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### What is a "based regular G-cover"

This is a simple question, but I can't seem to find the definition in the topology books I have lying around: What is a "based regular $G$-cover", where $G$ is a group? I am trying to ...
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### Projection of a Pentagonally-tiled Sphere

I know that a regular pentagonal tiling does not work in Euclidian space, but does work on a sphere. But this got me wondering something that I hope people can help with here, because I can't find any ...
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### Understanding proof of the Theorem 6.23 of the John Lee's Introduction to Riemannian Manifolds (about Riemannian covering map)

I am reading John Lee's Introduction to Riemannian Manifolds, Second Edition, proof of Theorem 6.12 and some questions arises. ( I think that I'm unfamilier to Riemannian Geometry, please ...
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### Does SO$(V, Q)$ have a unique connected double cover?

Let $V$ be a real or complex finite dimensional vector space with nondegenerate quadratic form $Q$. According to the spin representation Wikipedia article, Up to group isomorphism, SO$(V, Q)$ has a ...
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### What is the relationship between projective representation of a Lie group $G$, and an ordinary one of its universal cover?

What is the relationship between projective representation of a Lie group $G$, and an ordinary one of its universal cover if restricted to the finite-dimensional irreducible case? Whether an ordinary ...
1 vote
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### Problem 12-1 in Lee's Introduction to Topological Manifolds

Problem 12-1 from Introduction to Topological Manifolds by Lee: Suppose $q_1 : E \to X_1$ and $q_2 : E \to X_2$ are normal coverings. Show that there exists a covering $X_1 \to X_2$ making the ...
1 vote
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### Projecting a path homotopy "down" with a covering map. Do we also get a path homotpy?

Consider we have two spaces $X$ and $\tilde X$ and a covering map, $p\colon \tilde X \to X$. Let's say we have two paths $\tilde f$ and $\tilde g$ in $\tilde X$ that are lifts of $f$ and $g$, in $X.$ ...
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### Checking normality of a 4-sheeted covering of the Klein bottle by a torus

The figure gives a 4-sheeted covering map from the torus to the Klein bottle. I am trying to show that this covering map is not normal, but I got stuck. (Actually I am solving an exercise to construct ...
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### Is the universal cover of figure-8 contractible?

The universal cover of the figure-8 is the Cayley graph of the free group on $2$ generators with generating set $\{a,b\}$. So it is a tree. I know that finite trees are contractible. But this Cayley ...
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