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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

Example 3.13 Let Q 8 be

Example 3.13 Let Q 8 be the quaternion group. We denote by Z(Q 8 ) the centreof 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 conjugatesof 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 subgroupof 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

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