For questions about or involving covering spaces in algebraic topology.

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9
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1answer
369 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
5
votes
1answer
188 views

Liftings of curves $u\cdot v$ and $v\cdot u$ with respect to the sine covering map.

I'm trying to work through the exercises in Otto Forster's book on Riemann Surfaces. While most of them seemed not that hard, this one gives me a headache: Let $X=\mathbb{C}\setminus\{\pm1\}$ and $Y ...
0
votes
1answer
50 views

Covering maps are proper?

Under wich conditions a covering map is also proper? For example the covering of the circle is clearly not proper Is there anything more general that say, when the cover is a compact space? Or having ...
0
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0answers
25 views

Properties of loops under lifting

Let $p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0) $ be a covering space. Is it always true that if the image of a path $\tilde{\gamma}$ under $p_*$ is a loop $\gamma$ based at $x_0$, then ...
0
votes
3answers
32 views

Show that if $\rho$ is idempotent then $\rho$ acts as the identity on $\rho(V)$

A linear map $V \xrightarrow{\rho} V$ is idempotent if $\rho\rho = \rho$. Show that if $\rho$ is idempotent then $\rho$ acts as the identity on $\rho(V)$. (Such linear maps are called projections: ...
0
votes
2answers
32 views

Let $V \xrightarrow{\phi} W \xrightarrow{\psi} V$. Show that $\phi$ is injective and $\psi$ is surjective. [duplicate]

Let $V \xrightarrow{\phi} W \xrightarrow{\psi} V$ be linear maps such that $V\xrightarrow{\psi\phi}V$ is an isomorphism. Show that $\phi$ is injective and $\psi$ is surjective. So, I know that an ...
5
votes
2answers
574 views

Universal cover via paths vs. ad hoc constructions

I'm looking for some intuition regarding universal covers of topological spaces. $\textbf{Setup:}$ For a topological space $X$ with sufficient adjectives we can construct a/the simply connected ...
1
vote
2answers
53 views

Universal covering group and fundamental group of $SO(n)$

The universal cover of $SO(2)$ is $\mathbb{R}$, whilst the fundamental group is $\mathbb{Z}$. That is $$ SO(2) \cong \mathrm{universal\ cover}/\pi_1 $$ Likewise, I believe that the universal cover of ...
0
votes
1answer
33 views

If an open set is evenly covered by $p$, then its open subset is also evenly covered by $p$.

If $U$ is an open set evenly covered by $p: E\to B$ and $W$ is an open set contained in $U$, then $W$ is also evenly covered by $p$. I'm trying to prove this statement, but have a difficulty. So ...
0
votes
1answer
42 views

Understanding Lemma 54.2 in Munkres Topology

The image below (a Lemma and proof) is taken from Topology by James R. Munkres, 2nd Edition. Munkres Lemma 54.2 I understand the entirety of the proof up and till proving the statement: $$ F \text{ ...
1
vote
1answer
40 views

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

Let $G=\mathbb Z\rtimes\mathbb Z$ and $H=\mathbb Z\rtimes7\mathbb Z$. 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 \mid ...
0
votes
2answers
43 views

Is convergent or divergent $\sum_{n=1}^\infty{(-1)^n\dfrac{\ln{n}}{n\ln{\ln{n}}}}$?

$$\sum_{n=1}^\infty{(-1)^n\dfrac{\ln{n}}{n\ln{\ln{n}}}}$$ Any suggestions? I tried absolute convergence, but it doesn't work.
9
votes
3answers
131 views

Homotopy equivalent spaces have homotopy equivalent universal covers

A problem in section 1.3 of Hatcher's Algebraic Topology is Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. ...
2
votes
1answer
37 views

Show a function is continuous based on its properties relative to a covering map

Let $p:E\to B$ be a covering map, let $Y$ be locally path-connected, and let $g:Y\to E$ be a function such that $p\circ g$ is continuous $g \circ \gamma $ is continuous for every path $\gamma$ in ...
1
vote
1answer
78 views

Use of a Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified. The books gives no justification.
0
votes
0answers
27 views

Open Covers for Čech Homology

Would it make sense to define a filtered open covering $$\{U_i\} = U_1 < U_2 < \cdots < U_n = X$$ on a topological space $X$ in order to compute Čech homology? Or does this defeat the ...
0
votes
1answer
43 views

Covering space action

In exercise 1.3.28 in Hatcher's Algebraic Topology, we are asked to show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. This is a ...
1
vote
1answer
35 views

Deck transformations and compact CW complexes

Let $X$ be a CW complex and $\widetilde{X}$ its universal cover, formed by lifting the CW structure on $X$. A finite cellular cochain, denoted $\phi$, is a cochain in ...
4
votes
1answer
29 views

Deck transformations of cover of double mapping cylinder

On page 66 of Hatcher's Algebraic Topology, he discusses the universal cover of a space $X$ which is a cylinder with its edges glued to a circle by maps $z \mapsto z^m$ and $z \mapsto z^n$. He ...
1
vote
1answer
34 views

Classification of $G$-principal Bundle and classification of $G$-coverings: a bridge between the two?

I encountered the following sentence in an exercise (the context is irrelevant) Let $G\cong\langle s_1,s_2,\dots , s_g \mid R \rangle$ be a discrete one relator group. Consider the $G$-principal ...
0
votes
1answer
27 views

Covering group $Aut(\tilde{X},p)\cong NH/H $

I want to show $Aut(\tilde{X},p)\cong NH/H $ where $H=p_*\pi_1(\tilde{X},\tilde{x_0})$ and $NH$ is the normalizer of $H$. In my text book, the author sketches the proof of the above by using the ...
1
vote
1answer
35 views

Determine a normal covering space

Is there a way to determine if a covering space is normal without using the two theorems of Hatcher's book in pages 71 and 72?
0
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2answers
50 views

Suppose $p:E\to B$ is a covering map and $B$ is connected. Prove that if $p^{-1}(\{b\})$ has n points $p^{-1}(\{b\})$ has n points for every $b\in B$

My idea is to somehow show that the group $O_n$ is both open closed which will imply $O_n=B$. Then assign to each $n$ the set of points $O_n\subseteq B$ such that $p^{-1}(b)$ has exactly $n$ points. ...
6
votes
1answer
135 views

Universal cover of $T^2 \vee \mathbb{R}P^2 $

What is the universal cover of the wedge sum of the torus and the real projective plane? I know from Hatcher's Algebraic Topology that the universal cover of $\mathbb{R}P^2 \vee \mathbb{R}P^2 $ is ...
3
votes
0answers
43 views

Euler characteristic of 2-sheeted covering space

I'm currently taking a course on algebraic topology and while doing exercises, I realised that I wanted to use the following: If $X$ is a compact connected $2$-manifold and $\varpi:Y \rightarrow X ...
1
vote
1answer
60 views

The universal cover of a path-connected, locally path-connected space $X$ covers any other covering space

I'm currently reading Hatcher's Algebraic topology book. In page 68 he says: A consequence of the lifting criterion is that a simply-connected covering space of a path-connected, locally ...
1
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0answers
25 views

When is a fibration (canonically) a principal fibration over its group of automorphisms?

The question is inspired by the following observation: Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal ...
0
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0answers
28 views

Prove that covering maps are quotient maps

I know this to be true because my professor's lecture notes uses this result but I would like to see a proof of the statement.
4
votes
1answer
66 views

How can I prove that the hawaiian earring has no universal cover?

I know that the Hawaiian earring is not semi-locally simply connected so the existence is not guaranteed. Also, the point in which it must fail is the origin, where it isn't even locally simply ...
2
votes
0answers
31 views

Is there any general method to calculate the universal cover of a given topological space $X$?

I am currently taking a course in algebraic topology and I have to calculate the universal cover of a lot of spaces (I'll call them $X$ from here on). So far I know some tricks to do it: consider $X'$ ...
3
votes
1answer
64 views

Covering of hawaiian earring

I'm taking a course on Algebraic Topology and I'm struggling to find the solution to this problem: Let $Y$ be the Hawaiian earring in $\mathbb{R}^2$ and $Y'$ the union on infinite $Y$s moved $3z$ ...
0
votes
1answer
13 views

Example of non locally connected space with a covering in each connected component which is not a covering of the whole space

Can someone show me an example of an space $X$ non locally connected and another space $X'$ such that $\varpi: X' \to X$ is not a covering but $\varpi: \varpi^{-1}(X_i) \to X_i$ is, for each connected ...
2
votes
0answers
43 views

3 sheeted cover of Klein bottle with torus

So I'm dealing with this exercise in which it is asked to determine whether the torus can be a 3-cover of the Klein bottle. A friend of mine came up with a proof that this is not the case, but this ...
0
votes
1answer
18 views

Degree one branched cover is a homeomorphism

Suppose that $f:X \to Y$ is a branched cover of Riemann surfaces and a covering map of degree one outside of the ramification points. Then is $f$ a homeomorphism?
0
votes
1answer
26 views

The projection onto the orbit space $X/G$

Let $X$ be a locally compact, Hausdorff, path connected and locally path connected space. Assume a group $G$ acts freely and properly discontinuously on $X$, which means $\forall K^{compact},~~\{g\in ...
2
votes
2answers
43 views

How can I show that the composition of two coverings is also a covering?

I'm trying to prove the following: Let $\varpi ' : X'' \to X'$ and $\varpi : X' \to X$ be two coverings and let $X$ be a locally simply connected space. Prove that $\varpi \circ \varpi ' : X'' ...
1
vote
1answer
16 views

Definition of finite-sheeted covering

What is the definition of a finite-sheeted covering $q: E \to X$? Does it mean that every open $V \subseteq X$ has a pre-image $q^{-1}(V)$ that is the disjoint union of a finite number of sheets? Or ...
1
vote
1answer
110 views

let $p: E\rightarrow B$ continuous and surjective. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices is unique.

Let $p: E\rightarrow B$ coninuous and surjective and suppose that $U$ is an open set of $B$ that is evenly covered by p and U. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices ...
1
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0answers
18 views

Show entropy bound unit simplex

Let $\mathcal{S}_d$ be the $d$-dimensional unit simplex. Then for the norm $||x||_1 = \sum_i |x_i|$ and $0 < \varepsilon \leq 1$, $$N(\varepsilon, \mathcal{S}_d, ||\cdot||_1) \leq ...
1
vote
0answers
43 views

Find a nonregular 3-fold covering space of the genus two closed orientable surface.

Find a nonregular 3-fold covering space of the genus two closed orientable surface. This question was asked to me in Ph.D. Preliminary Exam. I have not any idea.
2
votes
1answer
57 views

On a subgroup of the deck transformation of a covering space

I'm stuck with an exercise. Suppose you have a covering space $M \rightarrow X$, and you define $G:=\{\tau \in Deck(M)|\tau(S)=S\}$, for some $2$-sphere $S$ in $M$, and $G$ acts freely by isometries ...
0
votes
0answers
22 views

Show covering number $N(\epsilon,\mathcal{P},h) < \infty$ for all $\epsilon >0$

Let $\mathcal{P} = \{P_{\theta}: \theta \in \Theta\}$ be a dominated model of distributions on $[0,1]$. For the parameter space $\Theta$ we have $$\Theta := \{\theta: [0,1] \rightarrow \mathbb{R} ...
0
votes
2answers
29 views

Size of the deck transformation group

If $p\colon Y\to X$ is a $k$-fold covering map, and $Y$ is path-connected, what is the size of Deck($p$), the deck transformation group? I was attempting to prove that the answer is $\leq k$, but ...
1
vote
0answers
30 views

Covering map of the annulus

How to find universal covering map of the annulus of inner radius $\frac{1}{R}$ and outer radius $R>1$ from the right half plane $H$ where $H=\{z|Re(z)>0\}$?
2
votes
1answer
30 views

Covering a rectangle with circles

On a rectangle table with area A, n unit-radius circles are placed and it is not possible to place any extra circles without overlapping with some of the existing ones or without placing circle's ...
2
votes
1answer
41 views

Surface groups and subgroups of fundamental groups

The fundamental group of any closed surface is a surface group. Let $S_3$ be the orientable surface of genus 3. Is $\pi_1(S_3)$ isomorphic to an index-3 subgroup of any surface group? We have 1 ...
2
votes
1answer
17 views

Covering space of an abelian topological group is abelian if the covering map is a homomorphism

I'm trying to show that if $(E, \cdot)$ and $(G, \cdot)$ are both topological groups, $G$ is abelian, and $(E, p)$ is a covering of $G$ such that $p:E\to G$ is a homomorphism with respect to $\cdot$, ...
1
vote
0answers
35 views

Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map.

Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map. I've almost completed solving this problem, but ...
3
votes
2answers
556 views

The fiber of a covering space over a connected space has constant cardinality

Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for every $b \in B$. I know that $E$ has ...
2
votes
1answer
32 views

Universal cover of boundary

Let $M$ be a compact manifold-with-boundary and $B$ a component of $\partial M$. Let $\tilde{M}$ be the univeral cover of $M$ with infinite-sheeted covering map $p:\tilde{M} \to M$. I wonder about the ...