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      Questions tagged [covering-spaces]

      The tag has no usage guidance.

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      Same fiber of induced covering map [closed]

      Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...
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      338 views

      Monodromy groups from Galois's viewpoint

      According to the Wikipedia article about monodromy, the monodromy group can be defined in terms of Galois theory in following way: Let $F(x)$ denote the field of the rational functions in the ...
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      236 views

      Finite etale covers of products of curves

      Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$. Let $C_1, C_2 \subset \mathbb{P}^1$ be non-empty ...
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      153 views

      Dyer–Lashof operations for more than 2 inputs

      Let $\mathcal{O}$ be a topological operad and $X$ an algebra over it. Let the base ring be $\mathbb{Z}_2$. If $C_*$ denotes the singular chain complex over $\mathbb{Z}_2$, the action of $\mathcal{O}$ ...
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      1answer
      216 views

      Covering with Deck group $\mathfrak{S}_3$

      I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be ...
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      1answer
      53 views

      Concerning the Spanier group relative to an open cover

      Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$?. Spanier defined $\pi (\mathcal{U}?, ?x)$ to be the subgroup of $\pi_1 (X?, ?x)$ which contains all homotopy classes having ...
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      85 views

      Singular homology: Lifting simplices gives map in homology

      Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$. Then the ...
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      46 views

      Galois Covering induces new Cover $Ind_H ^G(Y)$

      I have a question about the construction of the so called "induced cover" introduced in Tamas Szamuely's "Galois Groups and Fundamental Groups" (see page 84): We consider a group $G$ which contains a ...
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      votes
      1answer
      152 views

      English literature close to “Algébre et Théories Galoisiennes” by Régine and Adrien Douady

      I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...
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      1answer
      140 views

      Invariant lifts of a closed curve on a surface of genus > 1

      I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question : Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
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      2answers
      311 views

      Galois categories for topological spaces?

      Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)? ...
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      1answer
      215 views

      If $X, Y$ are topological spaces, with $Y$ being a k-space, and $f : X \to Y$ is a proper covering map, is $X$ necessarily a k-space?

      A k-space is a compactly generated Hausdorff topological space. (I used the terminology "k-space" in the question, in order keep the question within the limit of 150 characters.) Note that under the ...
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      93 views

      Idea behind definition of classifying space over an orbifold

      Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction. Definition : Let $X$ be a locally compact ...
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      1answer
      184 views

      Path-lifting property: function space interpretation

      I asked this question on math.SE, but even with a bounty, there were no answers/comments. I hope this is not too low-level for this site. Suppose I have a covering map $\pi:E\rightarrow B$, and a ...
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      2answers
      266 views

      covering theory with compact open topology

      In the following all spaces $C^0(X,Y)$ are spaces of base point preserving maps with the compact-open topology.Furthermore all spaces I consider in the following are locally pathwise connected. Under ...

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