Questions tagged [covering-spaces]
For questions about or involving covering spaces.
148 questions
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When does passing to universal covers preserve pushouts?
Given a span $X\stackrel{f}{\leftarrow} A \stackrel{g}{\to} Y$ of based path-connected topological spaces, we can form the pushout $X\cup_A Y$ as an adjunction space. (If one of the maps $f$ or $g$ is ...
- 37.6k
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Is the identity a universal covering of a contractible non-locally-path-connected space?
I was reading the chapter on covering space theory in Algebraic Topology by Tammo Tom Dieck.
The following problem is quite interesting (to me).
Context:
Given a principal $G$-covering $p : E \to B$, ...
- 81
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Does there exist an irregular simply-connected cover of a space with pathological local properties?
I have a question about covers of spaces with pathological local properties.
Let $(X,x_0)$ be a pointed path-connected space. Assume that there is a pointed covering space $p\colon (Y,y_0) \...
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4
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Connection between the monodromy group and the Galois group of a polynomial
Denote by $\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ the one-point compactification of the complex plane $\mathbb{C}$, and let $p:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ be a polynomial taken as a ...
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Finding "simple loops" on top of a ramified cover
Fix the following data, the $2$-sphere $S^2$ with a choise of basepoint $b \in S^2$ and for each $n$, fix punctures $x_1, \dots, x_n$ with choices of simple loops $\gamma_1, \dots, \gamma_n$ based at $...
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4
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Holomorphic Euler characteristic is multiplicative along covering maps: Is there a proof avoiding Hirzebruch–Riemann–Roch?
Let $M$ be a compact complex manifold, $p: \tilde{M} \to M$ a finite covering map (aka an étale cover) of order $m$. The Hirzebruch–Riemann–Roch theorem provides that
$$
χ(M) := \sum_p (-1)^p h^p(\...
- 1,333
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1
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Isotropic subgroup of Galois cover
Let $Y\rightarrow X$ be a ramified Galois cover between smooth algebraic curves. For a ramification point $y\in Y$, it is known that the isotropic subgroup $G_y$ is cyclic.
I am looking for a ...
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3
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141
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Reference Request: Pushforward of $\pi_1$ along a covering map and the Galois group
Let $f \colon (Y,y_0) \to (X,x_0)$ be a finite-to-one pointed covering map. The pushforward gives an inclusion $f_* \colon \pi_1(Y,y_0) \to \pi_1(X,x_0)$. If we take the universal cover $\widetilde{X}$...
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Descending universal branched cover
In Lawrence-Venkatesh, they tried to descend their construction of universal branched $G$-cover $Z^\circ\to Y^2-\Delta$ in Lemma 7.4. I have several questions about the proof.
They said the commuting ...
- 151
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2
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Analytic continuation gives a covering space (and not just a local homeomorphism)
Let $\mathcal{G}$ be the space of germs of holomorphic functions defined on open subsets of $\mathbb{C}$, topologized in the usual way. There is a natural map $p\colon \mathcal{G} \rightarrow \mathbb{...
- 111
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Finiteness of non-orientable 3-manifolds with the same orientable two-fold cover
Given a compact, orientable and boundary incompressible 3-manifold $M$. Suppose that either $M$ is closed, or $\partial M$ consists of tori.
For which non-orientable 3-manifolds $N$, the orientable ...
user519738
4
votes
1
answer
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Regular cover of Seifert fibered spaces with 3-exceptional fibers
I have a Seifert fibered 3-manifold with base manifold $S^2$ and 3 exceptional fibers, say $M(1/p, 1/q,1/r)$. Say $p,q,r$ are relatively prime. Is there an easy way to understand which Seifert fibered ...
7
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1
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What are the covering spaces of $\mathbb{Q}$?
Let $X = \mathbb{Q}$, topologized as a subset of the real line. Is there a reasonable description of the covering spaces of $X$?
Here is something more precise. One way of constructing covers $p: \...
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119
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For spaces $U$ and discrete sets $I,J$, are maps $f\colon U \times I \rightarrow U \times J$ commuting with the projection to $U$ covering spaces?
Let $U$ be a topological space, let $I$ and $J$ be discrete sets, and let $f\colon U \times I \rightarrow U \times J$ be a continuous map that commutes with the projection onto the first factor. In ...
- 405
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1
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Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?
Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
user519738