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      Questions tagged [covering-spaces]

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      Classify 2-fold covering spaces of three-times punctured plane [migrated]

      Ahoy Mathematicians, We are preparing for qualifying exams and ran across this question. There were two proposed methods of approaching it. Recognize that the thrice punctured plane is homotopic to ...
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      573 views

      Homology of the universal cover

      $k$ is a field. Let $X$ be a connected pointed $CW$-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous ...
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      2answers
      198 views

      $PSL_2(\mathbb{R})$ representations of free groups

      Let $S_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S_{g,n}^b$ deformation retracts to a bouquet ...
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      84 views

      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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      394 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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      252 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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      156 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
      227 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
      55 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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      86 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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      47 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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      1answer
      162 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
      143 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
      321 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)? ...
      3
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      1answer
      218 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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