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On Fundamental Groups of Galois Closures of Generic Projections

For the definition **of** ϑ(t a ) we have to check that for m, m ′ ≠ nf knm · t a · (f k nm ) −1 = f knm ′ · t a · (f knm ′ ) −1nmholds true. We conjugate by f ′ mm k and then we are done since t a and f ′kcommute by (†1).Hence the definition **of** ϑ does not depend on the choice **of** the m’s occuring.Next we want to show that ϑ defines an endomorphism **of** K ≤k−1n,d. For this wehave to show that the relations are preserved by ϑ. If we pick a relation from (∗1)to (∗4), (†1) and (†2) then we can find an index m distinct from the i, j, k, n’s inthis particular relation since we assumed n ≥ 5. The action **of** ϑ is then given byconjugating every element occuring in this relation by f nm k . Since the relationsform a normal subgroup this means that ϑ preserves the relations **of** K ≤k−1n,dandso ϑ defines an endomorphism **of** this group.Clearly, ϑ defines an automorphism **of** K ≤k−1n,dfor we can just define its inverseby replacing f nm k by (fk nm ) −1 in the definition **of** ϑ.To obtain K ≤kn,dfrom K≤k−1n,d∗ 〈t k 〉 we only need the relations (†1) and (†2) .For (†1) it is enough to consider all relations with a = k and arbitrary b:t k f b ij t k−1= f k nm f b ij (f k nm ) −1 = ϑ(f b ij ).We have to impose one relation for every m ≠ i, j but we have already shownabove that all these elements define the same element ϑ(f ij b ) **of** K ≤k−1n,d.And for a = k and b < k the relation (†2) is equivalent to−1t k t b t k = f nm k f nj b (f nm k ) −1 (f nj b ) −1 ·t} {{ } bapply (†4)= f mn k t b (f mn k ) −1 = ϑ(t b )As we have shown above this element does not depend on the choice **of** m ≠ n.Hence we have shown thatK ≤kn,d∼ =K≤k−1n,d∗ 〈t k 〉/ ≪ t k xt k −1 = ϑ(x) ∀x ∈ K ≤k−1n,d≫and this is precisely the semidirect product **of** K ≤k−1n,dby 〈t k 〉. □5.3 Affine subgroups and the construction **of** ˜K( − ,n)We denote by F d be the free group **of** rank d ≥ 1. We embed K(F d , n) as usualinto F d n , cf. Section 3.1.Definition 5.9 A subgroup **of** K(F d , n) with n ≥ 3 is called an affine subgroupif it is normally generated by elements **of** the form (r, r −1 , 1, ..., 1), r ∈ F d andtheir S n -conjugates.50

We note that for affine subgroups normal generation with respect to K(F d , n) hasthe same effect as normal generation with respect to F d n : This follows since weassumed n ≥ 3 and so we compute for (f, 1, f −1 , 1, ...) ∈ K(F d , n) and r ∈ F d :(f, 1, 1, ..., 1) (r, r −1 , 1, ..., 1) (f, 1, 1, ..., 1) −1= (f, 1, f −1 , ..., 1) (r, r −1 , 1, ..., 1) (f, 1, f −1 , ..., 1) −1 .We let G be a group and n ≥ 3 be a natural number. We then choose a presentationF d /N ∼ = G **of** G. Then we define R :=≪ K(N, n) ≫. This is an affine subgroup**of** K(F d , n) since it is normally generated by the elements (s, s −1 , 1, ..., 1) withs ∈ N and their S n -conjugates. We define˜K(G, n) := K(F d , n)/R.Since R is S n -invariant the S n -action on K(F d , n) descends to an action on thequotient ˜K(G, n) and we defineẼ(G, n) := ˜K(G, n) ⋊ S n .with respect to this action. This is well-defined because **of**Theorem 5.10 Let n ≥ 3 be a natural number. For every finitely generated groupG the construction **of** ˜K(G, n) and its S n -action do not depend on the choice **of** apresentation for G. Moreover, the construction **of** ˜K(−, n) is functorial in its firstargument.If we denote by H 2 (G) the second group homology **of** G with coefficients inthe integers then there is a central extension0 → H 2 (G) → ˜K(G, n) → K(G, n) → 1and the image **of** H 2 (G) lies inside the commutator subgroup **of** ˜K(G, n).PROOF. We embed K(F d , n) into F n nd . We denote by π the projection from F donto its last n − 1 factors. From Proposition 3.4 we know that ker π restricted toK(F d , n) equals the commutator subgroup [F d , F d ].We let f ∈ F d and s ∈ N. Then[(f, 1, f −1 , 1, ...), (s, s −1 , 1, ...)] = ([f, s], 1, 1, 1, ...)and this element lies in R. Thus [F d , N] is contained in R ∩ ker π.Conversely, R is generated by elements **of** the form (fsf −1 , s −1 , 1, ...) andtheir S n -conjugates where f runs through F d and s runs through N. From this itfollows that every element **of** R can be written as a product **of** the form∏(([fi , s i ], 1, ...) · (s i , 1, ..., s −1 i , 1, ...) ) .i(f i s i f i −1 , 1, ..., s i −1 , 1, ...) = ∏ i51

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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.

- 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 and 34: since the product over all componen
- 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: First, assume that i = n. ThenNow a
- 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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- 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
- Page 96 and 97: If we denote by Π g the fundamenta
- Page 98 and 99: NotationsVarieties and morphismsf :
- Page 100 and 101: [GR1][GR2][GH][SGA1]H. Grauert, R.