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 ...
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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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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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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 ...
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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 ...
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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 ...
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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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Euler characteristic of compact polyhedra. Is piecewise-linear map between compact polyhedra with constant fiber a covering?
$\newcommand{\cn}{\colon}$
Let $f\cn Y\to X$ be a piecewise-linear map between two compact
polyhedra. Suppose that the preimage of each point consists of
precisely $m$ points. Prove that $\chi(Y)=m\...
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Sufficient conditions of continous map to be covering maps
This is a question from "A First Course in Algebraic Topology" by C. Kosniowski
Let $p_1$: $X_1 \to X$, $p_2$: $X_2 \to X$ be covering maps with $X$ connected and locally path connected. (i) ...
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About covering spaces of the torus
Find the covering space of the torus $T=S^1\times S^1$ that is corresponding to the subgroup $2Z\oplus 3Z$ of $\pi_1(T,x_0)=Z^2$.
What is the covering space corresponding to the trivial group?, do one ...
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Show that the projection of a path along a meridian from south to north pole is non-trivial in real projective space for $n\geq2$
Let $\mathbb{R}\text{P}^n$ be the real projective space where antipodal points are identified and $n\geq2$. Let $\tilde{\gamma}:[0,1]\to \mathbb{S}^n$ be the path from the south pole $-P$ to the north ...
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Show that $\pi_1$ of this cover is a normal subgroup with index $\mathbb{Z}$
We are given a cover of the figure eight-
Show that $\pi_1$ of this cover is a normal subgroup with index $\mathbb{Z}$.
I have shown that the given cover is regular, hence the image of its ...
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Computing induced homomorphisms of covering maps over a punctured torus
Let $X$ be the space $S^1\times S^1 - \{p\}$ for a point $p$. Let $\pi:Y\to X$ be a covering map and let $\pi_*$ be an induced map on $\pi_1$ and $\pi_H$ be an induced map on $H_1$. Compute $\pi_*$ ...
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What is the universal covering of $\text{SL}_2(\mathbb{Z})\backslash\mathbb{H}$?
Let $\mathbb{H}$ be the upper half plane, and $\text{SL}_2(\mathbb{Z})$ acts on
$\mathbb{H}$ by linear transformation
$$\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}\cdot z = \frac{az+b}{cz+d}.$...
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Is $\operatorname{Spin}(n-1)$ a subgroup of ${\rm Spin}(n)$? [closed]
Just to double check that this statement, which looks true, is indeed true:
Is $\operatorname{Spin}(n-1)$ always a subgroup of $\operatorname{Spin}(n)$?
Any reference or quick argument would be ...
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Why is a finitely sheeted covering space of $\mathbb{R}^n$ never compact?
It is common to see the following consequence of Cartan-Hadamard stated as a significant result relating curvature and topology:
Let $M$ be a compact Riemannian manifold with non-positive sectional ...
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Understanding the "fundamental theorem of covering spaces"
I ask this question because I'm quite confused about the fundamental theorem of covering spaces ;
This theorem say that under suitable hypothesis for a topological space $X$ (semi locally simply ...
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Show that the index is 2 if and only if $\tilde{X}$ is path-connected. Show that the index is 1 if and only if $\tilde{X} \cong X\sqcup X$
Let $\pi\colon \tilde{X} \to X$ be a double cover, and let $X$ be path-connected. Show that the index is 2 if and only if $\tilde{X}$ is path-connected. Show that the index is 1 if and only if $\tilde{...
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Parameterizations of the circle $S^1$ and open arcs in $S^1$
The circle $S^1 = \{ z \in \mathbb C \mid \lvert z \rvert = 1\}$ inherits its standard topology from the plane $\mathbb C$ with the Euclidean topology.
The purpose of this question is to give a ...
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Each Preimage of an Evenly Covered Open set is a Component
Let $X$ and $Y$ be topological spaces, and let $q : X \to Y$ be a continuous map.
An open set $U \subseteq Y$ is said to be evenly covered by $q$ iff $q^{-1}[U]$ is a union of disjoint nonempty ...
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How to find first nontrivial fibration in Whitehead tower of the $n$-sphere?
Consider the Whitehead tower of the $n$-sphere $X = S^n$:
$$... \rightarrow X' \stackrel{p}{\rightarrow} X,$$
where $X'$ is $n$-connected. Explicitly, what is $X'$ and what is the fibration $p$? How ...
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Functor from orbit category to Top
Given a topological group $G$, define the orbit category $\mathcal{O}(G)$ as the category whose objects are the $G$-sets $G / H$, and whose morphisms are the $G$-equivariant maps. Peter May's book has ...
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Automorphism group is homeomorphic to quotient
I have been trying to prove that for any Riemann surface $X$ with universal cover $p: \tilde{X} \to X$, the group of automorphisms of $X$, $Aut(X)$, is isomorphic as a topological group (with the ...
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Do Lie algebras "know" about their Lie groups?
An undergraduate in physics asked me this question, and I did not know the answer, so I thought I would ask here. It is well-known that the Lie algebra to a Lie group is the tangent space to the ...
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Pullback of Etale Cover is Trivial
In Milne's notes on 'Etale Cohomology, Proposition 6.16 on page 49 is the following:
Proposition 6.16 Assume $X$ is connected, and let $\overline{x}$ be a geometric point of $X$. The map $\mathscr{F}\...
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Isomorphism of two different planar coverings in $S^2$
Let $G$ be a simple graph and $F$ be a closed surface. I will say that two embeddings $f_1 : G \to F$ and $f_2 : G \to F$ are equivalent if there is a homeomorphism $h$ of $F$ and an automorphism $\...
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The composition of covering maps between spaces that are compact and Hausdorff is a covering map
This is an exercise that I had to solve in a topology course.
Let X, Y, Z be Hausdorff compact spaces and let $p:X\rightarrow Y$ and$q:Y\rightarrow Z$ be covering maps. Show that $q\circ p:X\...
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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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Fundamental group of $3$-sheeted cover of Klein bottle
I'm currently in the middle of solving Hatcher Exercise 1.3.20
Construct nonnormal covering space of the Klein bottle by a Klein bottle and by a torus.
If $K$ is a Klein bottle, then $\pi_1(K)\simeq\...
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$p : E \to X$ is a connected, L.P.C. covering, show epimorphism $\pi_1(E)$ to covering transformations of $p$.
The following is an exercise from Greenberg and Harper's A First Course in Algebraic Topology.
Let $p : (E,e_0) \to (X,x_0)$ be a covering, with group of covering transformations $G$. If $E$ is ...
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