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On Fundamental Groups of Galois Closures of Generic Projections

Example 3.13 Let Q 8 be the quaternion group. We denote by Z(Q 8 ) the centre**of** Q 8 . Then the short exact sequenceinduces for all n ≥ 3 a sequence1 → Z(Q 8 ) → Q 8 → ¡ 2 × ¡ 2 → 11 → K(Z(Q 8 ), n) → K(Q 8 , n) → K(¡ 2 × ¡ 2, n) → 1that is not exact in the middle.We remark that K(Z(Q 8 ), n) is a normal subgroup **of** K(Q 8 , n).In particular, also the subgroup **of** K(Q 8 , n) generated by the conjugates**of** K(Z(Q 8 ), n) does not give the kernel **of** the surjective homomorphism fromK(Q 8 , n) onto K(¡ 2 × ¡ 2, n).PROOF. Using the presentation **of** Q 8 as in Example 3.12 we have Z(Q 8 ) =〈a 2 〉 = 〈b 2 〉. We can identify K(Z(Q 8 ), n) with the subgroup **of** Z(Q 8 ) n wherethe product over all components equals 1.Since Z(Q 8 ) is the centre **of** Q 8 we see that K(Z(Q 8 ), n) is a normal subgroup**of** Q n 8 and hence also a normal subgroup **of** K(Q 8 , n).The kernel **of** the surjective homomorphism K(Q 8 n)↠K(¡ , 2 ¡ × 2, n) equalsK(Q 8 , n) ∩ Z(Q 8 ) n . However, the element (a 2 , 1, ..., 1) lies in this kernel but notin K(Z(Q 8 ), n).□22

4 A first quotient **of** π 1 (X gal ) and π 1 (X affgal )Das Firmament blaut ewig, und die ErdeWird lange fest steh’n und aufblühn im Lenz.Du aber, Mensch, wie lang lebst denn du?Nicht hundert Jahre darfst du dich ergötzenAn all dem morschen Tande dieser Erde!4.1 Étale and topological fundamental groupsIn this section we recall some well-known facts that can be found e.g. in [SGA1,Exposé XII].Let X be an irreducible normal scheme **of** finite type over the complex numbersand let X an be its associated complex analytic space. Then we consider thefollowing three categories:1. The objects are connected and finite étale covers Y → X where Y is analgebraic scheme and the morphisms are morphisms **of** schemes over Xbetween these covers.2. The objects are connected holomorphic covers Y → X an where Y is acomplex space and the morphisms are holomorphic morphisms **of** complexspaces over X an between these covers.3. The objects are connected topological covers Y → X an where Y is a topologicalspace and the morphisms are continuous maps **of** topological spacesover X an between these covers.The relationship between these three categories is as follows:- Given a finite étale cover p : Y → X by a scheme Y this induces a finiteholomorphic cover p an : Y an → X an .Moreover, every algebraic morphism between finite étale covers **of** X inducesa unique holomorphic morphism between their analytifications.- Every holomorphic cover is also a topological cover and every holomorphicmap is continuous.- Every topological cover **of** X an can be given a unique structure **of** a complexspace such that the projection map onto X an becomes holomorphic.Moreover, every continuous map between holomorphic covers over X an canbe given a unique structure **of** a holomorphic morphism.23

- Page 1: On Fundamental GroupsofGalois Closu
- Page 5 and 6: ContentsIntroductioniii1 A short re
- Page 7 and 8: IntroductionSchon winkt der Wein im
- Page 9 and 10: is defined by a line bundle L on X.
- Page 11 and 12: depends only on G and n and not on
- Page 13 and 14: 1 A short reminder on fundamental g
- Page 15 and 16: Miyaoka [Mi] gave a construction of
- Page 17 and 18: 2 Generic projections and their Gal
- Page 19 and 20: 2.2 Galois closures of generic proj
- Page 21 and 22: Proposition 2.12 Let L be a suffici
- Page 23 and 24: 3 Semidirect products by symmetric
- Page 25 and 26: Remark 3.2 The assumption n ≥ 3 i
- Page 27 and 28: Corollary 3.5 Let n ≥ 2.1. If G i
- Page 29 and 30: The assertions about the torsion an
- Page 31 and 32: For j ≠ i, j ≥ 2 the group X τ
- Page 33: since the product over all componen
- Page 37 and 38: 4.2 The quotient for the étale fun
- Page 39 and 40: of K. The normalisation of X inside
- Page 41 and 42: Hence the induced homomorphism from
- Page 43 and 44: zero. Hence the proof also works al
- Page 45 and 46: an isomorphism between these two gr
- Page 47 and 48: In particular, for H = 1 we obtain
- Page 49 and 50: For an element g of the inertia gro
- Page 51 and 52: eThis automorphism has order e i an
- Page 53 and 54: Hence the homomorphisms from π top
- Page 55 and 56: 5 A generalised symmetric group5.1
- Page 57 and 58: Theorem 5.3 For n ≥ 5 there exist
- Page 59 and 60: Proposition 5.6 (Rowen, Teicher, Vi
- Page 61 and 62: First, assume that i = n. ThenNow a
- Page 63 and 64: We note that for affine subgroups n
- Page 65 and 66: Every element of H 2 (G) maps to an
- Page 67 and 68: PROOF. We let F d /N ∼ = G and F
- Page 69 and 70: 5.4 ExamplesWe now compute ˜K(−,
- Page 71 and 72: Theorem 5.22 For a K(G, 1)-complex
- Page 73 and 74: Example 5.24 The homomorphism ψ ma
- Page 75 and 76: 6 ConclusionJetzt nehmt den Wein! J
- Page 77 and 78: This can be also formulated as foll
- Page 79 and 80: Severi claimed that a curve with on
- Page 81 and 82: where ϕ denotes the splitting of
- Page 83 and 84: Corollary 6.3 Under the assumptions
- Page 85 and 86:
Proposition 6.5 Under the isomorphi

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Remark 6.8 Proposition 6.5 shows us

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We remark that the group on the lef

- Page 91:
E(π top1 (Z), n). By Corollary 3.3

- Page 94 and 95:
7.3 Surfaces in 3Let X m be a smoot

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If we denote by Π g the fundamenta

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NotationsVarieties and morphismsf :

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[GR1][GR2][GH][SGA1]H. Grauert, R.