For questions about or involving covering spaces in algebraic topology.

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1answer
63 views

Is it possible to have a connected manifold that is a double cover of a 2-sphere?

I have come up with a branched covering, but it necessarily has two branch points. From that I'm assuming that it can't be done, possibly related to the hairy ball theorem, but I don't know how to ...
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31 views

Non-Example of covering space

Let $X:= \mathbb{R_1} \sqcup \mathbb{R_2}$ where $\sqcup$ means the disjoint union. Now $X$ is a topological space with the "Disjoint topology", in which the opens are disjoint union of opens of $\...
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37 views

How to find fundamental groups and covering spaces of $\mathbb{RP}^2\vee \mathbb{RP}^2$?

The following is an exercise I was assigned in homotopy theory. Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$. a) Find $\pi_1(X)$. b) Find the universal cover of $X$. c) Find all of its connected $...
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1answer
48 views

Universal covering map and induced maps on homology and homotopy

I had an additional question regarding universal covering maps. If $p:U\rightarrow X$ is a universal covering map for space $X$ does it induce isomorphisms on homology $H_i$ for $i>1$. Or if that'...
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1answer
46 views

Showing there is no covering map $\mathbb{R}P^2\to T^2$

$\newcommand{\R}{\mathbb{R}}$ I'm being asked to show there is no covering map $\R P^2\to T^2$ (the torus) by showing that every map $\R P^2\to T^2$ is null-homotopic, by lifting to the universal ...
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1answer
63 views

Coverings of connected sum of four copies of $\mathbb{R}P^2$

G.Baumslag in one of his papers asserts that a group $G = \langle a,b,c,d | a^2b^2c^2d^2 = 1 \rangle$ contains all fundamental groups of closed compact orientable surfaces of genus $g\geq 2$? I think ...
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2answers
92 views

A detail about reconstructing covering space from the action $\pi_1(X,x_0)\to S_{p^{-1}(x_0)}$ in Hatcher's book

I'm struggling understanding a small sentece from Hatcher's Algebraic topology book (available online for free). In page 70 Hatcher wants to reconstruct the covering $p:\tilde X\to X$ from the ...
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1answer
41 views

When does triviality of $f_*$ imply null-homotopy of $f$?

I've been considering problems of the following type: Given certain topological spaces $X$ and $Y$ and a continuous function $f:X\to Y$, prove that $f$ is null-homotopic. The cases studied so ...
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1answer
26 views

Covering maps and Homotopy equivalence

I am struggling with the following fact: Say that we have $A\subset M$ a compact connected sumbanifold. Let $H\subset \pi_1(A)$ be a subgroup and $p_A:A_H\rightarrow A$ the corresponding covering ...
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1answer
61 views

The fundamental group of preimage of covering map

Define i: B $\to$ Y is an inclusion, p: X $\to$ Y is a covering map. Define $D=p^{-1}(B)$. We assume here B and Y are locally path-connected and semi-locally simply connected. Then if B,Y, X are path-...
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33 views

The extension of covering map

enter image description here If B,A,X are path connected, then in what relation of fundamental groups of B and A, the extension of q can be made?
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3answers
77 views

Fully cover a hyper-rectangle by two congruent disks ($n$-balls) and find the radius of them

I want to find a general way to calculate the smallest possible radius ($R$) of two congruent $n$-disks ($n$-balls) with the centers ($C_1$) and ($C_2$) lying on the diagonal of the hyper-rectangle ...
4
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1answer
39 views

Is a lift of a topological embedding still an embedding?

Let $p\colon E\to X$ be a covering map and $f\colon Z\to X$ a topological embedding. Suppose $F\colon Z\to E$ is a lift of $f$. Is $F$ still an embedding? What if I assume that $Z$ is connected and ...
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1answer
79 views

An example of lifting a group action to the universal cover.

Through a previous question, I understood how we can lift the action of a group $G$ on a topological space $X$ to an action of a covering group $G'$ of $G$ on the universal cover $\tilde{X}$ in such a ...
2
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1answer
57 views

Simply connected means universal-covering spaces

A covering space $p:(Y,y_0) \to (X,x_0)$ is called universal when $Y$ is simply connected (say that we restrict ourselves to path connected spaces, and locally path connected). I heard that this ...
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1answer
56 views

Is the quotient $X/G$ homeomorphic to $\tilde{X}/G'$?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
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19 views

A question from a proof about coverings given by orbits spaces

The proof is from here: About coverings given by orbit spaces I wonder why: $\pi_2\circ g = \iota \circ \pi_1$ is a different way of writing $H_2g \supset gH_1$
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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 $p^{-1}(x_0)...
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2answers
51 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
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1answer
40 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 ...
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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 $p^{-...
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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
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1answer
46 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
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1answer
70 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
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2answers
89 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 G-...
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1answer
64 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 ...
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2answers
58 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
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1answer
70 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 $G$ ...
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1answer
24 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 $\widetilde{c}$...
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1answer
66 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 ...
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0answers
39 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 $\tilde{\...
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3answers
38 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: $\...
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2answers
48 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 ...
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2answers
65 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
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1answer
53 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
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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{ ...
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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.
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0answers
30 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 ...
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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
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1answer
58 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 ...
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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 $H^n(\widetilde{X}^n,\widetilde{...
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249 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
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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 ...
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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 ...
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57 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 ...
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1answer
96 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 path-...
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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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32 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 ...
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1answer
175 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
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1answer
61 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 $X'$ ...