# Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

314 questions
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### Why do all the Platonic Solids exist?

In three dimensions it is quite easy to prove that there exist at most five Platonic Solids. Each has to have at least three polygons meeting at each vertex, and the angles of these polygons have to ...
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### How to prove that the center of the fundamental group of $T_g$ is trivial for $g \geq 2$?

Where $T_g$ is a closed orientable surface of genus g. I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization ...
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### Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
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### Given a curve over $\mathbb{P}^1$ how can I determine the monodromy around a ramification point?

Given a smooth curve of the form $$f(y) - g(x)$$ over $\mathbb{A}^1_x$ with a smooth compactification $C \to \mathbb{P}^1$, how can I determine the monodromy of the points of the fibers? Are there ...
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### When is the induced map $f^{*}:\operatorname{Spec}(S) \to \operatorname{Spec}(R)$ a covering map?

Let $f: R \to S$ be a ring homomorphism. When is the induced map $f^{*}: \operatorname{Spec}(S) \to \operatorname{Spec}(R)$ on topological spaces a covering map? More precisely, are there any ...
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### How can I find the monodromy of a cyclic galois cover of the affine line minus a few points?

Consider the following cyclic covering of the affine line minus a few points: $$\text{Spec}(\mathbb{C}[t,x]/(x^n - t(t-1)(t-2))) \to \mathbb{A}^1_t - \{ 0, 1, 2 \}$$ How can I find the local ...
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### Where can I find precise examples of ramified coverings of $\mathbb{C}P^1$?

One of the definitions of simple Hurwitz number $h_{g,\mu}$ is that it counts up to automorphisms the number of ramified coverings of $\mathbb{C}P^1$ such that covering space is a connected surface of ...
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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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### First Cohomology of Abelian Cover

Let $S$ be a closed oriented surface. Consider the (universal) abelian cover $p \colon S_{ab} \rightarrow S$, i.e. the one whose group of deck transformations is the abelianization of $\pi_1(S,\star)$....
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### Finite-sheeted covers of integer homology spheres

For an integer homology sphere $M$, one can show that non-trivial connected finite-sheeted covers of $M$ must have at least $5$ sheets. This is because, for a given $n$-sheeted cover $M'$, one can ...
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### 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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### How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
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### covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
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### Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
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### does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
Let $F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...