For questions about or involving covering spaces in algebraic topology.

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27 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
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1answer
36 views

Need to check if $H\triangleleft G$ in a covering of the klein bottle

Let $G=Z\times Z$ and $H=Z\times7Z$. I want to check if H◁G. I know I need to calculate N_G(H), and I think this is $$N_G(H)={(m,n) ∈ G | (m,n)(l,7k)(m,n)^{-1} ∈ H \forall l,k}$$ but I'm not sure ...
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2answers
86 views

$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
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1answer
41 views

For a covering map, if the target space is Hausdorff, so is the source

I am working on proving that if $q:E\rightarrow X$ is a covering map, and $X$ is Hausdorff, then so is $E$. The answer to this question: Domain is Hausdorff if image of covering map is Hausdorff ...
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2answers
46 views

Find all covering spaces of $\mathbb{RP}^n \times \mathbb{RP}^n$, $n>1$

Let $X = \mathbb{RP}^n \times \mathbb{RP}^n$. I know the following: the universal cover of $X$ is $Y = \Bbb S^n \times \Bbb S^n$ the fundamental group of $X$ is $G = \Bbb Z/2 \Bbb Z \times \Bbb Z/2 ...
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1answer
40 views

If $f\circ g$ is continuous and $f$ is a local homeomorphism, then $g$ is continuous

Suppose $g:X\to Y$ and $f:Y\to Z$, and $f$ is a local homeomorphism, which is to say that for any $y\in Y$ there is a neighborhood $U$ of $y$ such that $f\restriction U$ is a homeomorphism from $U$ to ...
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23 views

Automorphisms of simple covers of Riemann surfaces

Can anybody give me a simple proof that simple covers of a Riemann surface have no covering automorphisms?
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2answers
53 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
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1answer
64 views

Covering $S_2$ with $S_3$(or $S_n$)

How can I construct a covering map $p : S_3 \to S_2$? I can construct coverings for $S^1\vee S^1$, but same ideas don't work for $S_2$ ($S_g$ is sphere with $g$ handles).
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2answers
41 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
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3answers
49 views

simply connected covering of a path connected space (II)

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
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2answers
99 views

Finding the Fundamental Groups of Some Modular Spaces

I'm looking to compute the fundamental group of a couple of different quotients of the $n$-torus. The first of these I'm interested is the space $\mathbb{T}^n/S_n$ where the symmetric group $S_n$ ...
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0answers
28 views

Section of a covering projection from a connected space [duplicate]

Let $p:\overline{X}\rightarrow X$ is a continuous mapping. A continuous map $s:X\rightarrow \overline{X}$ such that $p\circ s =Id_X$ is called a section of $p$. Suppose $\overline{X}$ is connected ...
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1answer
31 views

if $V_1\cong U_1, V_2\cong U_2$, is $(V_1\cup V_2 \cong U_1\cup U_2)$? Pasting homeomorphisms

My question arises from the theory of covering spaces. assume $f:Y\to X$ is a covering map, or more generally a local homeomorphism. Assum $U_1,U_2\subset X$ are open sets such that $f|_{V_1}, ...
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93 views

A problem on covering space from Hatcher book…

I was trying a problem from Hatcher's book Algebraic Topology, in section 1.3 problem number 12. Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. ...
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1answer
73 views

simply connected covering of a path connected space

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
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1answer
31 views

About covering spaces

Suppose X is a topological space whose fundamental group is Z x Z x Z2 x Z3. Is it possible for the wedge sum of two circles to be a covering space for X? Can anyone help me with this ?
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1answer
46 views

Prove that a non-empty subset of an open set which is evenly covered is evenly covered

Let $p: E\rightarrow B$ a continuous surjective map and $U \subseteq B$ be open and not empty and who is being evenly covered by $p$. Show that all non-empty subsets of $U$ are being evenly covered by ...
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1answer
85 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
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3answers
42 views

locally path connectedness

While studying covering spaces , hatcher mentioned the "shrinking wedge of circles" this space is locally path connected as I was told , but I wasn't able to prove it nor to see it, it looks like comb ...
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1answer
76 views

Universal covering of the complement of a circle in $\mathbb{R}^3$

What is the universal covering of $X=\mathbb{R}^3\setminus(S^1\times\{0\})$? I've been trying to build a covering map from $\mathbb{R}^3$ onto $X$ via composition of $p:\mathbb{R}^3\to Y$ and $q:Y\to ...
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1answer
119 views

Algorithm - Circle Overlapping

Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape ...
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2answers
112 views

Finite fundamental group and covering spaces

Show that if a path-connected, locally path-connected space X has a finite fundamental group , then every map $X$ to $S^1 \times S^1$ is nullhomotopic (i.e. homotopic to a constant map) . Is the ...
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1answer
63 views

Constructing a simply connected covering space

"Construct a simply connected covering space of the space that is the union of the sphere S2 with two of its intersecting diameters." can anyone help me with this? i don't know how to think , all ...
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2answers
73 views

Explain why the following statement is false

Let $f:S^1 \to S^1$ be given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$. Then there is a unique lift $\bar f: S^1 \to \mathbb{R}$ with the properties that (i) $\bar f(1)=0$ and ...
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1answer
47 views

12.16 in Lee's Introduction To Topological Manifolds

Reading through Lee's Introduction To Topological Manifolds. Theorem 12.16 says the following: Suppose G and H are connected, locally path-connected topological groups, and $\phi:G \to H$ is a ...
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1answer
33 views

Is the tangent bundle of a covered manifold a quotient manifold?

Given a covering manifold $\rho :\widetilde M \to M$ we know that $M$ can be thought of as the quotient space of $\widetilde M$ like so $M = \widetilde M /\ G$ where $G$ is the monodromy group (or ...
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4answers
159 views

Longest chord in the intersection n disks (circle areas)

Given n disks that intersect, there is a shape in the space where they intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
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70 views

Is the Riemann surface for the square root simply connected?

I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...
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About prime geodesic cycles and deck transformations group

I'm proving theorem 2 occurring in Sunada's paper Riemannian coverings and isospectral manifolds. Unfortunately Sunada's quotes himself to the following paper: Tchbotarev’s density theorem for closed ...
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41 views

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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22 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
18 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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35 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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68 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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29 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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19 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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32 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
53 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
86 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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54 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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35 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
58 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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36 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 ...
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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
28 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
41 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 ...
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1answer
36 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 ...