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      Questions tagged [fundamental-group]

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      First thoughts about fundamental group of a topological (Lie) groupoid

      I am reading the paper Chern-Weil map for principal bundles over groupoids. In page number $13$, authors say let us recall the definition of fundamental group of a topological groupoid. But, they ...
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      2answers
      243 views

      Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid

      Let $X$ be a path-connected smooth manifold, it is known that: $$H^1(X):=H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R).$$ Explicitly, a closed one-form $\alpha$ gives a function on $\pi_1(X)$ by $[\...
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      1answer
      397 views

      Categorical Significance of Fibrations

      It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
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      110 views

      Fundamental groups of open algebraic varieties [closed]

      Let X be an algebraic variety over $\mathbb C$. 1. Is it possible to compute its fundamental group? 2. If X is two dimensional, what is its fundamental group? 3. Let $X\to \bar X$ be the inclusion to ...
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      1answer
      188 views

      The (topological) fundamental group of (quasi)-projective algebraic varieties

      I would like to know: What does the fundamental group of a quasi-projective algebraic variety look like? I remember that I have seen somewhere that for a connected, finite-type CW-complex $X$, ...
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      2answers
      257 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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      votes
      2answers
      383 views

      Fundamental group of a topological group

      It is well known that the fundamental group of a path-connected topological group is abelian. Suppose that $G$ is a connected topological group and let $Ab(G)$ the abelianization of the topological ...
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      0answers
      127 views

      Recover an etale fundamental group from local fundamental groups?

      Suppose we have a scheme $S$ and an etale covering $\{ U_i \to S \}$. How much information about $\pi_1^{et} (S)$ can we recover from all the $\pi_1^{et} (U_i)$ ? Are there special conditions we can ...
      3
      votes
      1answer
      57 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 ...
      3
      votes
      1answer
      144 views

      Approximation of homotopy avoiding a point in $\mathbb{R}^3$

      For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\...
      4
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      1answer
      114 views

      Geodesic representatives in the orbifold fundamental group

      Does every element in the orbifold fundamental group $\pi_1^{orb}(X,x)$ of a closed hyperbolic 2-orbifold $X$ admit a unique geodesic arc representing it? Does every free homotopy class in $X$ admit ...
      4
      votes
      1answer
      145 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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      0answers
      254 views

      Relationships among constructions of fundamental group for schemes

      There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
      5
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      2answers
      332 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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      0answers
      482 views

      What would be the simplest analog of Langlands in algebraic topology?

      It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...

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