For questions about or involving covering spaces in algebraic topology.

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degree of $f\circ g$

Let $f,g : \mathbb{S}^1 \to \mathbb{S}^1$ be two continuous maps where $\mathbb{S}^1$ is the unit circle. Prove that $\deg (f \circ g) = \deg f \deg g$. I don't know at all how to do it : I first set ...
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How to lift maps going into the base of a fiber bundle?

Let $p:E\to B$ be a fiber bundle and $f:B'\to B$ a map. Under what conditions does a lift $f':B'\to E$ exist? In the context of covering spaces, I remember a necessary and sufficient condition is that ...
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Universal cover of a CW complex corresponding to an identification space

I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the ...
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19 views

Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
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Cover a polygon with polygons

Besides right angled triangles, is there any polygon I could use to cover any given (regular or not) polygon? It's clear that given a triangle, square, hexagon or rectangle you would other options. ...
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18 views

Which is correct between cover or covering?

Let $I$ be closed n-dimensional intervals, $$I=\{\mathbf{x}: a_j\le x_j \le b_j, \quad j=1, \cdots, n\}$$ and $S$ be a countable collection of intervals $I_k$, $$S=\{I_j,\quad j=1, 2, \cdots\}$$ In ...
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Proof of the quotient map $\pi : X\to X/G$ is a covering map only if the action of $G$ is properly discontinuous.

The following is a theorem from Munkres' Topology and there's a part in the proof of the theorem that I don't understand. I've written the part in bold. $X/G$ is the orbit space obtained from $X$ by ...
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Maximal analytic continuation gives rise to a covering

Suppose that $a$ is a point on a connected Riemann surface $X$ and $\varphi \in \mathcal{O}_a$ admits an analytic continuation along every curve in $X$ starting at $a$. Let $(Y, p, f, b)$ be the ...
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Branch points of $f : \mathbb{C} \to \mathbb{P}^1 : z \mapsto \frac{1}{2}(z + \frac{1}{z})$

Problem: find the branch points of the function $$ f : \mathbb{C} \to \mathbb{P}^1 : z \mapsto \frac{1}{2}\bigg(z + \frac{1}{z}\bigg). $$ My try: The zeros are $i$ and $-i$, but I don't see why $f|V$ ...
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25 views

Materials on some construction involving classification of covering spaces

Let $X_u \rightarrow X $ be a universal covering. Let $S $ be any set with a group $\pi(X,x_0) $ acting on it from the right side. Then we get space $S \times X_u $ with a group action $ S \times X_u \...
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89 views

Field $K$ with $\operatorname{Gal}(\overline{K}/K)\simeq\widehat{F_2}$

Is there a field K such that $\operatorname{Gal}(\overline{K}/K)$ is the profinite free group with two generators? For one generator I know that for all the $\mathbb{F}_p$ we have $\operatorname{Gal}(...
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Construction of K(G,1) for arbitrary group G

In Example 1B.7 of Hatcher's 'Algebraic Topology', he attempts to construct a $K(G,1)$ space for any group G. His construction goes as follows: Consider the $\Delta$-complex (denote it as $EG$) ...
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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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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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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
50 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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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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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
43 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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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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62 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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34 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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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 ...
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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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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 ...
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59 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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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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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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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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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 ...
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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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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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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 ...
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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 ...
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71 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 $\textbf{unique path lifting}$ ...
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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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70 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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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 ...
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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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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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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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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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40 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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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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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 $...
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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 ...
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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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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.