For questions about or involving covering spaces in algebraic topology.

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4
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2answers
60 views

Does $\pi_1(X, x_0)$ act on $\tilde{X}$?

Let $X$ be a path connected, locally path connected space and let $p:\tilde{X} \to X$ be a covering map. Let $x_0\in X$. Then we have a natural right action of $\pi_1(X, x_0)$ on the fibre ...
2
votes
2answers
50 views

A technical detail in the construction of a covering space: Is the projection $p$ continuous without assuming semilocal simple connectedness?

I am studying the construction of a covering space with prescribed characteristic subgroup. For simplicity, I will outline the case where the characteristic subgroup is trivial (i.e. we're dealing ...
0
votes
1answer
36 views

universal cover of a fiber bundle.

In the case of a cartesian product of two (nice enough) topological spaces $X\times Y$ it is known that the universal cover is the cartesian product of the universal covers of the factors. In the more ...
3
votes
0answers
43 views

Do all paths have a neighborhood about them which lift homeomorphically to a covering space?

Let $\widetilde{X}$ be a covering space of $X$ with projection map $p:\widetilde{X}\to X$. We know by definition that, for all $x$ in $X$, there exists a neighborhood $x\in U\subset X$ such that ...
0
votes
1answer
23 views

Covering Projections and Quotients

Let $Y$ be a covering space of $X$, where $p:Y\to X$ is a covering projection. For $x\in X$, define the fiber of $x$ as $p^{-1}(x)$. Set up an equivalence relation on $Y$ as $y_1\sim y_2$ if they are ...
2
votes
1answer
45 views

Topological Groups and Covering Spaces

So the question is suppose $G$ is a topological group and $H$ is a closed, discrete subgroup of $G$, we have to show that the quotient map $p: G\to \frac GH$ is a covering projection. The way I'm ...
2
votes
1answer
61 views

Covering Space Counter example

Given a covering space $(p,\tilde{X})$ of a space $X$, we know that every covering application $p:\tilde{X} \rightarrow X$ is a local homeomorphism and possesses the $\textbf{unique path lifting}$ ...
3
votes
2answers
85 views

Classifying Covering Spaces using First Cohomology

I am familiar with the classification of covering spaces of a space $X$ in terms of subgroups of $\pi_1(X)$ (up to conjugation). However, if $X$ is a manifold, I know that $H^1(X; G)$ classifies ...
0
votes
1answer
62 views

Covering map in the context of Riemann Surfaces and Algebraic Topology

I am taking a course in Riemann surfaces and our lecturer has warned us that the definition of covering maps in the context of Riemann surfaces is strictly weaker than the ones used in Algebraic ...
1
vote
2answers
48 views

Algorithm to cover maximal number of points with one circle of given radius

we have a plane with some points on it. We know coordinate of each point apriori. We also have a circle of unit radius. I need an algorithm that determines optimal/sub-optimal position of a circle ...
3
votes
1answer
69 views

If the Lie algebra is ${\frak g}={\frak a}\oplus{\frak b}$ then the Lie group is $G=AB$?

Let $G$ be a connected Lie group and suppose that its Lie algebra ${\frak g}$ splits into a direct sum of ideals $${\frak g} = {\frak a}\oplus{\frak b}.$$ Let $A$ be the connected Lie subgroup of $G$ ...
1
vote
1answer
22 views

Is the transfer homomorphism in cohomology surjective?

Let $p:\widetilde{X} \to X$ be an $n$-sheeted covering space. Consider a singular simplex $c:\triangle^k \to X$, because the simplex is simply-connected, there exist $n$ different lifts ...
0
votes
1answer
57 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
votes
0answers
38 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
37 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
43 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 ...
1
vote
2answers
63 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
50 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
45 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{ ...
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.
0
votes
0answers
29 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 ...
2
votes
1answer
39 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 ...
0
votes
1answer
55 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
38 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 ...
9
votes
3answers
236 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 $Y$. ...
4
votes
1answer
33 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
40 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 ...
3
votes
0answers
53 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
91 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 ...
0
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0answers
29 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.
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0answers
31 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 ...
6
votes
1answer
169 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 ...
2
votes
1answer
60 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 $X'$ ...
4
votes
1answer
92 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 ...
3
votes
1answer
85 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
15 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
62 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
22 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
30 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 ...
0
votes
1answer
29 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
42 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 ...
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0answers
28 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
47 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
2answers
44 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'' ...
2
votes
1answer
58 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
27 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} ...
1
vote
1answer
39 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?
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0answers
34 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 $H=\{z|Re(z)>0\}$?
0
votes
2answers
49 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 ...
3
votes
1answer
56 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 ...