Fundamental group of Galois covers of degree 5 surfaces

This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is [Formula: see text] where c = gcd (a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface F1(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1(2, 2) with respect to a generic projection is isomorphic to [Formula: see text].

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Sur l'existence du sch\'ema en groupes fondametal

Épijournal de Géométrie Algébrique ◽

10.46298/epiga.2020.volume4.5436 ◽

2020 ◽

Vol Volume 4 ◽

Author(s):

Marco Antei ◽

Michel Emsalem ◽

Carlo Gasbarri

Keyword(s):

Fundamental Group ◽

Finite Type ◽

Group Scheme ◽

Final Version ◽

Finite Fundamental Group ◽

Galois Covers ◽

The One

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group. Comment: in French. Final version (finally!)

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Fundamental group of Galois covers of degree 6 surfaces

Journal of Topology and Analysis ◽

10.1142/s1793525321500412 ◽

2021 ◽

pp. 1-21

Author(s):

M. Amram ◽

C. Gong ◽

U. Sinichkin ◽

S.-L. Tan ◽

W.-Y. Xu ◽

...

Keyword(s):

Fundamental Group ◽

Algebraic Surfaces ◽

Fundamental Groups ◽

Galois Cover ◽

Chern Numbers ◽

Galois Covers

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings.

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Filling words in fundamental groups of Riemann surfaces

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.1.1227 ◽

2013 ◽

Vol 50 (1) ◽

pp. 31-50

Author(s):

C. Zhang

Keyword(s):

Riemann Surface ◽

Fundamental Group ◽

Riemann Surfaces ◽

Fundamental Groups ◽

Closed Geodesics ◽

New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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Representations of the fundamental group and the torsor of deformations. Global aspects

10.23943/princeton/9780691170282.003.0003 ◽

2017 ◽

Author(s):

Ahmed Abbes ◽

Michel Gros

Keyword(s):

Fundamental Group ◽

Koszul Complex ◽

Finiteness Conditions ◽

Inverse Systems ◽

Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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On the fundamental group of the complement of a complex hyperplane arrangement

Arrangements – Tokyo 1998 ◽

10.2969/aspm/02710257 ◽

2018 ◽

Cited By ~ 4

Author(s):

Luis Paris

Keyword(s):

Fundamental Group ◽

Hyperplane Arrangement ◽

Complex Hyperplane

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TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

Glasgow Mathematical Journal ◽

10.1017/s0017089521000215 ◽

2021 ◽

pp. 1-8

Author(s):

DANIEL KASPROWSKI ◽

MARKUS LAND

Keyword(s):

Fundamental Group ◽

Homotopy Equivalent ◽

Poincaré Duality ◽

Poincare Duality ◽

Canonical Map ◽

Duality Space ◽

Simply Connected Manifolds ◽

Simply Connected ◽

Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

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