For questions about or involving covering spaces in algebraic topology.

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Covering number of the set of $n_1\times n_2$ matrices of rank at most $r$

What is the covering number of the set of $n_1\times n_2$ matrices of rank at most $r$? We know that the dimension of the set is $r(n_1+n_2-r)$. Thus, the covering number $N(\rho)\le C ...
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covering number of cartesian product of manifolds

Let there be two manifolds $P$ and $Q$, with their covering numbers $N_P(\rho)$ and $N_Q(\rho$), respectively. Is it true that the covering number for the Cartesian Product $P\times Q$, $N_{P\times ...
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Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
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limit of covering spaces

Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, ...
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Deck group of a connected n-fold cover must have at most n elements

Let $p:Y\to X$ be an $n$-fold covering map, with $Y$ connected. Show that $Deck(p)$ has at most $n$ elements. My thinking was to prove this by contradiction, i.e. suppose we have distinct ...
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Fundamental group of a covering space

I understand the correspondence between the subgroups of a fundamental group $\pi_1(X)$ and the covering spaces of $X$. However, I do not understand what is implied about the fundamental groups of ...
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38 views

Is the Fundamental Group of space with contractible universal cover torsion free?

Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle $K$ torsion free? We have the following presentation: $$\pi_1(K) = \langle a,b: aba = b ...
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328 views

On double covering of projective plane and map preserving antipodal points

Let $p:S^2\to P^2$ be the double covering of the real projective plane. Let $g:P^2 \to P^2$ be a map such that its induced homomorphism on fundamental group is not trivial. I'd like to show that ...
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45 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 ...
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37 views

With justification, determine whether or not the following space is compact.

The space in question is the Hausdorff topological space with base β: β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} . (I have confirmed that this in fact a base of a Hausdorff ...
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61 views

Two lifts of a local homeomorphism

Just learning about sheaves. Suppose I have a sheaf $\mathscr{F}$ on a topological space. (The sheaf can take values in sets, let's say.) Let $E \overset{\pi}{\to}X$ be the etale space of this ...
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397 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 ...
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19 views

Möbius strip covering space

how can we describe the universal covering space of the Möbius strip? the Möbius strip is a square $[0,1]\times [0,1]$ with identifications $(0,y)\sim (1,1-y)$. So my guess is that the universal ...
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Is this covering group $G'$ of $G$ unique?

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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what's the universal covering group of general linear group $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$

Because the general linear group is not simply connected, they must be able to be covered by some simply connected Lie group. But I cannot imagine which Lie group with same dimension can cover ...
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Fundamental group and universal cover for this quotient space

For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more ...
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Covering spaces of wedge sum of circles

Let's suppose I'd like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $ S^{1} \vee S^{2} $ it is easy to point out exactly how do all ...
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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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1answer
20 views

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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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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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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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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71 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 ...
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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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86 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 ...
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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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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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121 views

Universal covering and double cover functors

Cross-posted on MO Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know ...
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639 views

covering spaces and the fundamental groupoid

Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between ...
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124 views

Fundamental group and path-connected

Let $p:E \rightarrow B$ be a covering space, $E$ and $B$ are path-connected. Let $A$ be a path-connected subset of $B$ . How to use fundamental group to give a sufficient and necessary condition to ...
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54 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 ...
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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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60 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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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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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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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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42 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 ...
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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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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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Geodesic Lines on Covering Maps

So I'm not sure how deck transformations work into this problem. I've established the following so far. Let $\pi:\tilde{M}\rightarrow M$ be the universal covering map. We may suppose that $M$ is ...
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630 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
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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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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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28 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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49 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 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 ...