# Tagged Questions

For questions about or involving covering spaces in algebraic topology.

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### Branched coverings of the Riemann sphere

Can someone give me an example of a non-trivial branched covering of the Riemann sphere? Is there some way to enumerate all such coverings? Is there any easy answer to the same questions about the ...
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### Classification of Galois covering maps over a bouquet of 2 circles

The b-sheeted Galois covering maps over $C^*$ are equivalent to $z\mapsto z^b$. I wonder if there is an analogous statement for such Galois covers over C except two points $0,1$. Is that true that ...
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### Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps). Let $f:M^m\to N^m$ be a smooth finite covering map. -- The following implication is not true: $M$ can be embedded ...
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### If $G$ is a finite nontrivial group, then $K(G, 1)$ cannot be a finite CW-complex?

I am currently working on this algebraic topology problem and got stuck: Suppose $X$ is a finite CW complex with $\pi_1(X)$ a nontrivial finite group. Show that its universal cover $\widetilde{X}$ ...
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### A space $X$ with $S^{2n+1}$ as universal covering space must be orientable.

Problem If $X$ has $S^{2n+1}$ as universal covering space, then show that $X$ must be orientable. My idea: By contradiction, suppose $X$ is non-oreintable. Then we consider the orientation covering ...
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### Existence of the universal covering space of a connected Lie group

I am working on a project about how the universal cover of a connected Lie group is a Lie group, but I cannot find a theorem that assures that this universal cover actually exists. I've found ...
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### A space homeomorphic to the connected sum $\mathbb{RP}^3$ # $\mathbb{RP}^3$

Problem (1) Consider the space $Y$ obtained from $S^2 \times [0,1]$ by identifying $(x,0)$ with $(-x,0)$ and also identifying $(x,1)$ with $(-x,1)$, for all $x\in S^2$.Show that $Y$ is ...
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### Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
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### Covering spaces of $S^1$

Put $\tilde X=\lbrace (exp(2\pi if(t)),t)| t\in \mathbb{R} \rbrace$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is any continuous function and let $\pi_1$ be the projecction on the first coordinate. ...
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### Isometry algebra implication from Riemannian covering

I really wish that, if $\pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h})$ is a Riemannian covering, then $\mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g})$, where $\mathfrak{i}(M,\mathrm{g})$...
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### Show that if $f$ is a proper surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is well ...
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### An exhaustive continuous map is a covering map.

$p_1:\tilde X_1 \rightarrow X \, ; \, p_2:\tilde X_2 \rightarrow X$ two coverings maps, where $X$ connected and locally path-connected, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
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### Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
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### 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 (m,n)(l,7k)(m,n)^{-...
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### If $f\circ g$ is continuous and $f$ is a local homeomorphism, then $g$ is continuous

Suppose $g:X\to Y$ and $f:Y\to Z$, and $f$ is a local homeomorphism, which is to say that for any $y\in Y$ there is a neighborhood $U$ of $y$ such that $f\restriction U$ is a homeomorphism from $U$ to ...
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### Automorphisms of simple covers of Riemann surfaces

Can anybody give me a simple proof that simple covers of a Riemann surface have no covering automorphisms?
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### Regular and non-regular covering spaces of $\Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1}$.

I tried to draw the regular and non-regular covering spaces of $\Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1}$. I think the regular covering space is: Is it true? How do you draw the non-regular ...
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### Covering $S_2$ with $S_3$(or $S_n$)

How can I construct a covering map $p : S_3 \to S_2$? I can construct coverings for $S^1\vee S^1$, but same ideas don't work for $S_2$ ($S_g$ is sphere with $g$ handles).
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### Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
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### A problem on covering space from Hatcher book…

I was trying a problem from Hatcher's book Algebraic Topology, in section 1.3 problem number 12. Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. ...
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### Algorithm - Circle Overlapping

Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape ...
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### Finite fundamental group and covering spaces

Show that if a path-connected, locally path-connected space X has a finite fundamental group , then every map $X$ to $S^1 \times S^1$ is nullhomotopic (i.e. homotopic to a constant map) . Is the ...
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### Constructing a simply connected covering space

"Construct a simply connected covering space of the space that is the union of the sphere S2 with two of its intersecting diameters." can anyone help me with this? i don't know how to think , all ...
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### Explain why the following statement is false

Let $f:S^1 \to S^1$ be given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$. Then there is a unique lift $\bar f: S^1 \to \mathbb{R}$ with the properties that (i) $\bar f(1)=0$ and (ii)...
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### 12.16 in Lee's Introduction To Topological Manifolds

Reading through Lee's Introduction To Topological Manifolds. Theorem 12.16 says the following: Suppose G and H are connected, locally path-connected topological groups, and $\phi:G \to H$ is a ...
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### Is the tangent bundle of a covered manifold a quotient manifold?

Given a covering manifold $\rho :\widetilde M \to M$ we know that $M$ can be thought of as the quotient space of $\widetilde M$ like so $M = \widetilde M /\ G$ where $G$ is the monodromy group (or ...
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### Longest chord in the intersection n disks (circle areas)

Given n disks that intersect, there is a shape in the space where they intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...