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On Fundamental Groups of Galois Closures of Generic Projections

Examples 5.21 The following H 2 ’s vanishH 2 (¡ n) = H 2 (Q 8 ) = H 2 (¡ ) = H 2 (D ∞ ) = 1,where Q 8 denotes the quaternion group and D ∞ denotes the infinite dihedralgroup. For the dihedral groups **of** order 2n we have{1 if n is oddH 2 (D 2n ) =2 if n ¡ is even.For n ≥ 4 it is known thatAppendix to Section 5H 2 (¡ 2 × ¡ 2) = H 2 (S n ) = ¡ 2.5.5 Group homology and the computation **of** H 2In this section we first recall the construction **of** group homology. Then we givesome **of** its properties and give some statements that allow us to actually computeH 2 **of** a given group. As references we refer to [Br, Chapter II], [We, Chapter 6],[Rot, Chapter 7] and [Rot, Chapter 11].Let G be an arbitrary group. For a left G-module M we define its module **of**co-invariants to be the quotient **of** M by the module I G generated by all elementsg · m − m for all g ∈ G and m ∈ M:M G := M/I G .Taking co-invariants defines a right exact functor for left G-modules and we canconsider its left derived functor. We define the i.th homology H i (G) **of** G to be thei.th left derived functor **of** − G applied to the G-module ¡ with trivial G-action:H i (G) := H i (G, ¡ ).Using the standard resolution **of** ¡ over the group ring ¡ [G] it is not hard to provethat for all groupsH 0 (G) ∼ = ¡H 1 (G) ∼ = Gabholds true. Clearly, all homology groups are abelian groups. Using again thestandard resolution mentioned before one can show that if G is a finite group thenalso its homology groups are finite.The origins **of** group homology lie in algebraic topology: We recall that aconnected CW-complex Y is called a K(G, 1)-complex if π top1 (Y ) ∼ = G and if itsuniversal cover is contractible.58

Theorem 5.22 For a K(G, 1)-complex Y there exist for all i ≥ 0 isomorphismsH i (G) ∼ = Hi (Y, ¡ )where H i (Y, ¡ ) denotes the singular homology **of** the topological space Y .A short exact sequence0 → A → X → G → 1is called a central extension **of** G if A lies in the centre **of** X. A central extension0 → A → X → G → 1 is called a universal central extension if for every centralextension 1 → B → Y → G → 1 there exists a unique homomorphism from Xto Y making the following diagram commute0 → A → X → G → 1↓ ↓ ||0 → B → Y → G → 1If such a universal central extension exists it is unique up to isomorphism.Central extensions 0 → A → X → G → 1 with a fixed abelian group Aare classified by Hom(H 2 (G), A). In particular, central extensions with ¨ ∗ areclassified by Hom(H 2 (G), ¨ ∗ ) ∼ = H 2 (G, ¨ ∗ ) =: M(G). This latter group iscalled the Schur multiplier **of** G. If G is finite then Pontryagin duality provides uswith a non-canonical isomorphism between H 2 (G) and M(G).A group G has a universal central extension ˜G if and only if it is perfect. Inthis case the universal extension takes the form0 → H 2 (G) → ˜G → G → 1.Now let N be a normal subgroup **of** a free group F such that G ∼ = F/N. Thenthere is a central extension0 → (N ∩ [F, F ])/[N, F ] → [F, F ]/[N, F ] → [G, G] → 1.In case G is a perfect group this is exactly its universal central extension. But evenin the case where G is not necessarily perfect we have the followingTheorem 5.23 (Hopf) Let G be an arbitrary group. If N is a normal subgroup**of** a free group F such that G ∼ = F/N thenH 2 (G) ∼ = (N ∩ [F, F ])/[F, N].59

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On Fundamental GroupsofGalois Closu

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ContentsIntroductioniii1 A short re

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IntroductionSchon winkt der Wein im

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is defined by a line bundle L on X.

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depends only on G and n and not on

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1 A short reminder on fundamental g

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Miyaoka [Mi] gave a construction of

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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
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- Page 35 and 36: 4 A first quotient of π 1 (X gal )
- 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
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- 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
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- Page 98 and 99: NotationsVarieties and morphismsf :
- Page 100 and 101: [GR1][GR2][GH][SGA1]H. Grauert, R.