For questions about or involving covering spaces in algebraic topology.

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universal cover homotopy equivalent if the base space homotopy equivalent

I am working on Hatcher's algebraic topology book and I got stuck in problem 8 in section 1.3. It says if $\hat{X}$ and $\hat{Y}$ are simply-connected covering space of the path connected, locally ...
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24 views

Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
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1answer
19 views

Understanding a proof of lifting $F:Y\times I\rightarrow X$ to $\widetilde F:Y\times I\rightarrow \widetilde X$

The statement to prove given in Allen Hatcher's book Algebraic Topology is: Given a map $F:Y\times I\rightarrow X$ and a map $\widetilde F:Y\times \{0\}\rightarrow \widetilde X$ lifting ...
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40 views

How to define multiplication in covering group?

Let $G$ be a connected topological group and let $p:\tilde{G}\to G$ be a universal covering of $G$. Then $\tilde{G}$ is also a topological group and $p$ is a continuous homomorphism. My question is: ...
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72 views

Covering between universal covers

While trying to solve a problem, an intuitive idea has brought me to the following statement. Is it true? If yes, how can we prove it? If $X$ is a covering space of $Y$, then the universal cover of ...
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38 views

Uniqueness for a covering map lift: is locally connected necessary?

So I just got through proving the following theorem: If $p:C\to X$ is a covering map and $Y$ is a [xxx] space, then given $y_0\in Y$, $c_0\in C$, $f:Y\to X$ such that $f(y_0)=p(c_0)$ there exists ...
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24 views

Lifting property of a covering map, product topology version

Suppose I have the following theorem (1): If $C,X$ are spaces, $p:C\to X$ is a covering map, $Y$ is a "nice" topological space (I think simply connected and locally path-connected is sufficient), ...
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33 views

Is there a construction on covering spaces that yields the free product on fundamental groups?

$\DeclareMathOperator{\Aut}{Aut}$ Suppose $p_1 \colon E_1 \to B_1$ and $p_2 \colon E_2 \to B_2$ are regular covering maps, with corresponding group exact sequences $1 \to \pi_1(E_i) \to \pi_1(B_i) \to ...
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2answers
32 views

Fibers in Covering Spaces

While reading Hatcher, he stated "If $p: \tilde{X} \rightarrow X$ is a covering space, then the cardinality of the set $p^{-1}(x)$ is locally constant. I have trouble seeing that this is the case. I ...
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1answer
62 views

About the definition of universal covering space

There are some references (for instance in Greenberg & Harper) that consider the universal covering space to be not only simply connected but also locally path connected. This definition seems to ...
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2answers
99 views

Universal covering space of wedge sum

Consider the wedge sum of the unit circle and real projective plane $S^{1} \vee \mathbb{R}P^{2}$. How would one construct a universal covering space for this kind of wege sum? I've tried constructing ...
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1answer
60 views

about path connected covering spaces.

Let $p:E\rightarrow X$ be a covering space. It is well known that if $X$ is connected, then all the fibers have the same cardinality. This can be seen as a simple consequence of the fact that the ...
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0answers
37 views

Morphism induced in cohomology of a covering space

It is a basic question but I'm stuck. If $p:M\rightarrow N$ is a $m$-fold unramified covering between surfaces, why the morphism induced by $p$ in cohomology at level 2 with coefficients in ...
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1answer
64 views

How to find universal covering space?

If our topological space is connected, locally connected and semi-locally simply-connected, then we know that a universal cover exists. Knowing the existence, my question is how to find universal ...
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1answer
29 views

Covering Space of Triangulable Space

Assuming a triangulable space is one homeomorphic to a simplical complex. How can one prove that any covering space of a triangulable space is triangulable? I know that one can lift the ...
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47 views

How to find n-sheeted covering space of a topological space explicitly(If exists)?

We know that there is a one to one Galois correspondence between subgroups of the fundamental group of some topological space and covering space for a path connected and locally path connected ...
2
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1answer
72 views

what is the two sheeted covering space of a sphere with a diameter?

I have calculated the fundamental group of sphere with a diameter using Van-Kampen theorem, which is $Z$. So corresponding to subgroup $2Z$ there exist a two sheeted connected covering space. So ...
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1answer
32 views

Classification of $S^1$ coverings

How do I prove that every covering map of $S^1$ is isomorphic to one of the following covering maps? $$\varepsilon: \mathbb{R} \to S^1, \quad z\mapsto e^{2\pi i z} \\ p_n:S^1 \to S^1, \quad z \mapsto ...
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1answer
48 views

Show that the covering space of a smooth manifold is a smooth manifold.

I indeed found this question Is a covering space of a manifold always a manifold. However I do not know the concepts here used. As far as I know I just need to present a suitable atlas for the ...
2
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1answer
41 views

4-fold regular coverings of wedge of two circles

Problem 7 (p.80) of Algebraic Topology book by Tammo Tom asks to classify all 4-fold regular coverings of a wedge of two circles. I am aware that the required coverings correspond to normal subgroups ...
4
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2answers
123 views

Alternate construction of the universal cover of a space

Suppose you have a connected, locally path connected Hausdorff space $Y$ that admits a universal covering (i.e. is semilocally simply connected). It occured to me that maybe one can describe the ...
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19 views

which distance function is better to use

I have large data sets with large features space. I'm hesitating between finding the distance between each of those data sets to cluster them into 4 or 5 clusters. or just apply a method by using a ...
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1answer
40 views

If $p:E\to X$ is a covering map ($X$ connected and locally arcwise connected) then is $E$ locally connected?

I recall my definition of a covering map. A continuous and surjective map $p:E\to X$ between topological space, where $X$ is connected and locally arcwise-connected, is called a covering map if for ...
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1answer
57 views

Is this plus sign a covering space of $S^1 \vee S^1$?

Consider this space, where open circles denote missing endpoints: Hatcher (p57) says that "every covering space of $S^1\vee S^1$ is a [2-oriented] graph." The above space is not a graph since the ...
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26 views

injectivity of the map covering the inclusion $SO(n)\subset GL^+(n)$

Let $n\ge 2$ and $\theta\colon Spin(n)\rightarrow SO(n,\mathbb{R})$ be the two-fold covering of $SO(n,\mathbb{R})$ by the spin group $Spin(n)$, $\tilde{\theta}\colon ...
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1answer
60 views

Universal covering space of $S_{2}/\sim$, where $\sim$ is certain relation.

Let $p,q$ be different points of $S_{2}=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}=1\}$. We consider the space $X=S_{2}/\sim$ where $\sim$ is the next relation: $x,y\in S_{2}$, $x \sim y$ if and ...
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44 views

Surjective map from orientable covering space to orientation cover of base space.

Let $p \colon M \to N$ be a covering space, and let $M,N$ be manifolds. Assume now that $M$ is orientable and $N$ is not orientable. I'm asked to find a covering map $q \colon M \to N$ s.t. $p= \pi ...
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2answers
71 views

Converse of the path lifting lemma

We know that covering spaces have the path lifting property i.e. if $p:E \rightarrow B$ is a covering map and $u:I \rightarrow B$ is a path wish intial point $a$, then for each $w \in p^{-1}(a)$, ...
3
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1answer
48 views

If the fibers of a quotient map are all discrete, is this map a covering map?

If $p:\tilde{X}\rightarrow X$ is a covering projection then I know that for every point $x \in X$ the fibre above $x$, i.e $p^{-1}(x)$, has the discrete topology. Here $p$ being a covering map means ...
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1answer
41 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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140 views

Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
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2answers
52 views

Induced subgroup of $\pi_1(S^1)$ by $p_n$

Consider the following covering map $p_n: S^1 \to S^1, z \mapsto z^n$. Why is the subgroup of $\pi_1(S^1)$ induced by $p_n$ isomorphic to $n\mathbb{Z}$? I know that $\pi_1(S^1) \cong \mathbb{Z}$ but ...
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0answers
32 views

Covering Number of Lipschitz Function Space

On page 12 of the slide deck here, the author gives an example where a lower bound (and upper bound, but I am particularly interested in the lower bound) on the $\epsilon$-covering number of a ...
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38 views

Coverings maps of a simply connected space

Let be $Y$ a simply connected space. Show that $Y$ doesn't admit covering maps that aren't homeomorphisms, ie, every cover space of $Y$ is trivial ($I\times Y$, with $I$ a discrete space). So, I know ...
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2answers
37 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $f^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
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111 views

the upper bound of covering $\{46,26,2\}$ or $\{47,26,2\}$

As to $\{v,k,2\}$ covering, can you show the upper bound of $A(46,26,2) = 6$ or the upper bound of $A(47,26,2) = 6$? See also Photo Booth problem
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49 views

Correspondence between first homology group and deck transformations.

Let $M$ be a connected topological manifold with universal covering $\pi: \widetilde{M} \rightarrow M$ and let $p \in \widetilde{M}$ be a point. Let $\alpha,\beta : \widetilde{M} \rightarrow ...
2
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1answer
70 views

Topological covering + local diffeomorphism gives smooth covering

I got stuck at some point while working on this part of an exercise from Lee's Introduction to Smooth Manifolds, 2nd edition. The part which I am stuck on is to prove (one of the directions of ...
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0answers
17 views

Construction of the completion of a spread

I am reading Fox's paper Covering Spaces with Singularities and am a bit confused in a part where he constructs completions of spreads. The relevant definitions may be found here: A Complete Spread ...
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1answer
74 views

Is this a covering space of $S^1 \vee S^1$?

Is the following a covering space of $S^1 \vee S^1$ ? It would appear so since there is no point that has more than 2 incoming or outgoing arrows. It seems that the potential covering map $p:Y\to ...
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2answers
82 views

Covering space of $S^1 \vee S^1$?

Is this a covering space of $S^1 \vee S^1$? I'm not sure what the map from this space onto $S^1 \vee S^1$ does. What is mapped onto which $S^1$?
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0answers
91 views

Unique Path lifting of covering map

Let $p:E\rightarrow B$ be a covering map (in particular $p$ is a fiber bundle with discrete fiber). We want to prove the following: Given a commuting diagram of the following form: $\{0\}\rightarrow ...
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2answers
89 views

Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...
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0answers
60 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
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1answer
58 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
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70 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
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1answer
53 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
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0answers
41 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
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1answer
36 views

Show that the free group on $n$ generators is a finite index subgroup of $F_2$

Using covering spaces, prove that for each integer $n \geq 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators. I get how the cayley graph of $F_n$ would be ...
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1answer
39 views

Based covering maps for a bouquet of two circles

For each of the following subgroups of $$ \left \langle x,y \right \rangle = \pi_{1}(S^{1}\vee S^{1}) $$ construct a based covering map $$ \ p:(\tilde{X},\tilde{b})\rightarrow (S^{1}\vee S^{1},b) ...