The fundamental group of a $\mathcal{CP}^2$ complement of a branch curve as an extension of a solvable group by a symmetric group

1999 ◽  
Vol 314 (1) ◽  
pp. 19-38 ◽  
Author(s):  
Mina Teicher
Keyword(s):  
Fundamental Group ◽  
Symmetric Group ◽  
Solvable Group ◽  
Branch Curve

THE FUNDAMENTAL GROUP OF THE GALOIS COVER OF HIRZEBRUCH SURFACE F1(2, 2)

2007 ◽  
Vol 17 (03) ◽  
pp. 507-525 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Hirzebruch Surface ◽  
Generic Projection ◽  
Galois Cover ◽  
Hirzebruch Surfaces ◽  
Galois Covers ◽  
Branch Curve

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].

Representing 3-manifolds by a universal branching set

1983 ◽  
Vol 94 (1) ◽  
pp. 109-123 ◽  
Author(s):  
José María Montesinos
Keyword(s):  
Fundamental Group ◽  
Symmetric Group ◽  
Boundary Component ◽  
Heegaard Diagram

In this paper all 3-manifolds will be supposed to be compact, connected, oriented and without 2-spheres in the boundary.Given a 3-manifold M we obtain a closed pseudomanifold M^ by capping off each boundary component of M with a cone. We prove that such an M^ is a covering of S3 branched over a subcomplex G of S3 which is independent of M, and such that S3 - G has free fundamental group on two generators. Hence M^ (and also M) can be represented by a transitive pair {σ, τ} of permutations in the symmetric group Σh on the set {1,2, …, h}, for some h. We show how to obtain {σ, τ} from a given Heegaard diagram of M.

Linear configurations of complete graphs K4 and K5 in ℝ3, and higher dimensional analogs

2017 ◽  
Vol 26 (05) ◽  
pp. 1750028
Author(s):  
Andrew Marshall
Keyword(s):  
Fundamental Group ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Simplicial Complexes ◽  
Alternating Group ◽  
Complete Graphs ◽  
Mapping Cylinder ◽  
Higher Dimensional ◽  
Special Cases

We investigate the space [Formula: see text] of images of linearly embedded finite simplicial complexes in [Formula: see text] isomorphic to a given complex [Formula: see text], focusing on two special cases: [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, and [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, so [Formula: see text] has codimension 2 in [Formula: see text], in both cases. The main result is that for [Formula: see text], [Formula: see text] (for either [Formula: see text]) deformation retracts to a subspace homeomorphic to the double mapping cylinder [Formula: see text] where [Formula: see text] is the alternating group and [Formula: see text] the symmetric group. The resulting fundamental group provides an example of a generalization of the braid group, which is the fundamental group of the configuration space of points in the plane.

The fundamental group of the complement of the branch curve of ℂℙ1 × T

2009 ◽  
Vol 25 (9) ◽  
pp. 1443-1458 ◽  
Author(s):  
Meirav Amram ◽  
Michael Friedman ◽  
Mina Teicher
Keyword(s):  
Fundamental Group ◽  
Branch Curve

scholarly journals The fundamental group of the complement of the branch curve of the second Hirzebruch surface

Topology ◽  
2009 ◽  
Vol 48 (1) ◽  
pp. 23-40 ◽  
Author(s):  
Meirav Amram ◽  
Michael Friedman ◽  
Mina Teicher
Keyword(s):  
Fundamental Group ◽  
Hirzebruch Surface ◽  
Branch Curve

scholarly journals The fundamental group of the Quillen complex of the symmetric group

2004 ◽  
Vol 282 (1) ◽  
pp. 33-57 ◽  
Author(s):  
Rached Ksontini
Keyword(s):  
Fundamental Group ◽  
Symmetric Group

On the Branch Points in the Branched Coverings of Links

1985 ◽  
Vol 28 (2) ◽  
pp. 165-173
Author(s):  
Shintchi Kinoshita
Keyword(s):  
Fundamental Group ◽  
Symmetric Group ◽  
Branch Points ◽  
Branched Coverings ◽  
Branched Covering ◽  
Exact Position ◽  
Monodromy Map

AbstractLet l be a polygonal link in a 3-sphere S3 and a branched covering of l, which depends on the choice of a monodromy map ϕ. Let be the link in over l. In this paper we determine the exact position of in for some cases. For instance, if l is a torus link ((n + 1)p, n) and ϕ is an appropriate monodromy map of the fundamental group of S3 - l into the symmetric group of degree n + 1, then is an S3 and l is a torus link (np,n2). The 3-fold irregular branched covering of a doubled knot k is an S3, if it exists. The position of the link over k is shown in a figure. The link over knot 61 is obtained by K. A. Perko and the author, independently, and shown without proof in a paper by R. H. Fox [Can. J. Math. 22 (1970), 193-201]. The result mentioned in the above includes this case.

Representations of the Infinite Symmetric Group

2016 ◽  
Author(s):  
Alexei Borodin ◽  
Grigori Olshanski
Keyword(s):  
Symmetric Group ◽  
Infinite Symmetric Group
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