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      Questions tagged [fundamental-group]

      The tag has no usage guidance.

      5
      votes
      2answers
      235 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 ...
      6
      votes
      2answers
      356 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 ...
      3
      votes
      0answers
      125 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
      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 ...
      3
      votes
      1answer
      136 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
      votes
      1answer
      91 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
      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 ...
      8
      votes
      0answers
      249 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
      votes
      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)? ...
      15
      votes
      0answers
      455 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 ...
      3
      votes
      1answer
      107 views

      Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial

      I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative). If we wish to construct a 1-variable polynomial $A(t)$, we ...
      19
      votes
      2answers
      611 views

      Can one compute the fundamental group of a complex variety? Other topological invariants? [duplicate]

      Given a system of polynomial equations with rational coefficients, is there an algorithm to compute the geometric fundamental group of the variety defined by these equations? I'm interested in both ...
      3
      votes
      1answer
      391 views

      A projective (or free) $\mathbb{Z}\pi_1$-module

      Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces ...
      6
      votes
      1answer
      117 views

      Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$

      The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...
      8
      votes
      0answers
      162 views

      Fundamental group of moduli space of K3's

      According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...

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