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Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

Let $\pi : E \to B$ be a continuous surjective map between topological spaces $E$ and $B$. We say that $\pi$ is a covering map if for every $x \in B$, there is an open neighbourhood $U$ of $x$ such that $\pi^{-1}(U)$ is a union of disjoint open sets in $E$, each of which is mapped homeomorphically onto $U$ by $\pi$.

We call $E$ a covering space of $B$ and often refer to $B$ as the base space.

The open neighbourhoods referred to in the definition are often called evenly covered neighbourhoods.

The fibres of $\pi$ are homeomorphic, so they all have the same cardinality; this cardinality is often called the number of sheets of the covering.

Reference: Covering space.

1696 questions
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A good way to understand Galois covering?

A covering map $f:X\rightarrow Y$ is called Galois if for each $y\in Y$ and each pair of lifts $x, x^{'}$, there is a covering transformation taking $x$ to $x^{'}$. What is a good way to understand this definition? It seems to me that $f$ is Galois…
M. K.
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Monodromy correspondence

Lately I've been studying monodromy and covering maps (in particular ramified covering mapos of Riemann surfaces), and I came across something I didn't fully understand. Let $V$ be a connected real manifold, and let $\rho:\pi_1(V,q)\to S_d$ be a…
Robert Auffarth
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To find all connected 6-fold cover of $\mathbb R P^2 \# \mathbb R P^2 \# \mathbb R P^2 $

What is the full list of connected 6-fold covers of $\mathbb R P^2 \# \mathbb R P^2 \# \mathbb R P^2$, up to covering isomorphism? This is equivalent to finding all subgroups $H$ of $G = \langle a,\ b,\ c \ | \ aabbcc = 1 \rangle$ such that…
Springfield
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Prove that there exist a regular covering of the 8 figure of degree at most $n!$

Let $\phi : Y \rightarrow X$ be a $n$-sheet covering of the 8 figure. Prove that there exist $\psi : Z \rightarrow X$ an at most $n!$- sheet regular covering such that the diagram commutes (where $\mu$ is a covering). $\require{AMScd}$ \begin{CD} Z…
GingFreecss17
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Lifting an automorphism to the universal covering space..

Let $X$ be a manifold and $Y$ be its universal covering. Is it true that any $\phi \in \mathrm{Aut}(X)$ can be lifted to $\overline{\phi}\in \mathrm{Aut}(Y)$?
M. K.
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Factorisation of covering maps

It’s a well-known fact that for continuous functions $f:X\to Y$ and $g:Y\to Z$ of locally connected spaces, if $g\circ f$ and $g$ are covering maps, so is $f$. I was wondering if the following statement is also true: If $g\circ f$ and $f$ are…
FKranhold
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How morphism of coverings induces morphism of automorphisms groups?

Let $X_1\to Y$ and $X_2\to Y$ be two coverings. How does a morphism $X_1\to X_2$ over $Y$ induce morphism $\operatorname{Aut}_Y(X_1)\to \operatorname{Aut}_Y(X_2)$? It should be trivial, but I can not understand it even in case of $Y$ point: if…
evgeny
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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 $pr_1:X\times F\to X$ is the projection. Suppose…
ALe0
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Recovering the lattice of subgroups of a fixed fundamental group

Let $X$ be locally path connected and semi-locally simply connected topological space. The fundamental theorem of covering spaces says the local system functor functor taking a covering space to the path-lifting action of its fundamental groupoid on…
Arrow
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A continuous and injective map $f: \mathbb{S}^2 \rightarrow \mathbb{S}^2$ is a homeomorphism

I found this question talking about covering spaces so I am trying to prove that f is a covering map. This give me the surjectivity I need to prove the question since the sphere is compact and $T_2$. Any idea?
davidivadful
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How do we construct the covering space of a lattice?

I am studying the concept of covering spaces in algebraic topology, and I am interested in understanding how to construct the covering space of a lattice. A lattice is a partially ordered set in which every pair of elements has a least upper bound…
Olandelie
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Discontinuous lifts of continuous maps

While recapitulating the theory of covering maps, I thought that it would a most inconvenient thing if the lift of a continuous map defined by continuation along paths to a covering space should be discontinuous under certain…
Cloudscape
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Universal covering space via fiber product

Given 3 topological spaces $X,Y,Z $ and 2 functions $ f:X \rightarrow Z $, $ g:Y \rightarrow Z$, I define the fiber product between $ X $ and $ Y $ over $ Z $ by: $$ X\times_Z Y := \{(x,y) \in X \times Y | f(x) = g(y) \}. $$ My problem is to find…
Michele Antonello
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Covering space of disconnected spaces

Let $f:M\to N$ is a $d-$fold covering. Is it true that $M$ is disconnected if $N$ is disconnected? If not, can anyone give a counter example?
usr1988
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Covering map of the torus boundary onto itself, lifting from the projective space

I'm trying to prove this rather simple fact that the following mapping is a covering and to check it's rank $p\colon S^1\times S^1\rightarrow S^1\times S^1$; $p(z_1,z_2)=(z_1^2,z_2^3).$ So I think that rank is 6 - $p^{-1}(a,b)$, for every two $z_1$…
whereabouts
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