- Text
- Generic,
- Subgroup,
- Fundamental,
- Element,
- Proposition,
- Generated,
- Homomorphism,
- Finite,
- Quotient,
- Projection,
- Galois,
- Closures,
- Projections

On Fundamental Groups of Galois Closures of Generic Projections

quotient does not depend on the choice **of** the presentation and we may refer toboth quotients as ˜K(G, n).Since R is S n -invariant the action **of** S n on K(F d , n) descends to the quotientK(F d , n)/R. A similar reasoning as above shows that also this action onlydepends on G and n.□Again we denote by p 1 the projection from F d n onto its first factor. By abuse **of**notation we will also denote its restriction to K(F d , n) with p 1 . As a consequence**of** the previous theorem we can determine quotients by affine subgroups:Corollary 5.11 Suppose we are a given a natural number n ≥ 3 and an affinesubgroup R **of** K(F d , n). We defineThen there is an isomorphismN := p 1 (R) and G := F d /N.K(F d , n)/R ∼ = ˜K(G, n).In particular, the quotient is completely determined by G and n.PROOF. Since p 1 is surjective the subgroup N **of** F d is indeed normal. Also R isstable under S n and so N does not depend on the projection we have chosen.We have a short exact sequence1 → N n ∩ K(F d , n) → K(F d , n) → K(G, n) → 1.Clearly K(N, n) is a subgroup **of** R and since R is a normal subgroup also its normalclosure with respect to K(F d , n) is contained in R. Conversely, R is normallygenerated by elements **of** the form (r, r −1 , 1, ...) and their S n -conjugates. Sincethese r’s lie in N we conclude that R must be contained in ≪ K(N, n) ≫ and soR and ≪ K(N, n) ≫ coincide. Hence K(F d , n)/R is isomorphic to ˜K(G, n) bydefinition **of** the latter group.□Corollary 5.12 If α : G → H is a homomorphism between finitely generatedgroups then there are induced maps0 → H 2 (G) → ˜K(G, n) → K(G, n) → 1↓ ↓ ↓0 → H 2 (H) → ˜K(H, n) → K(H, n) → 1 .The induced map K(G, n) → K(H, n) coincides with the one induced fromK(−, n). The map from H 2 (G) to H 2 (H) can be made compatible with the mapinduced from group homology.54

PROOF. We let F d /N ∼ = G and F d ′/N ′ ∼ = H be presentations **of** G and H,respectively. Again we lift α : G → H to a map ϕ : F d → F d ′. The map betweenthe two H 2 ’s is the one induced from ϕ andH 2 (G) ∼ = (N ∩ [Fd , F d ])/[F d , N]↓H 2 (H) ∼ = (N ′ ∩ [F d ′, F d ′])/[F d ′, N ′ ]By [Br, Exercise II.6.3.b] this can be made compatible with the homomorphismα ∗ : H 2 (G) → H 2 (H) on homology.□The connection with the universality results for K(−, n) given in Proposition3.9 and Corollary 3.10 is as follows:Corollary 5.13 Let n ≥ 3 be a natural number and G be a finitely generatedgroup. With respect to the action **of** S n on ˜K(G, n) given by Theorem 5.10 wedefineX := ˜K(G, n) and Y := XSn /X (1) S.n−1Then Y is isomorphic to G and X is equal to X Sn . The universal homomorphismgiven by Proposition 3.9 takes the following form:1 → ⋂ ni=1 X → XS (i)Sn → K(Y, n) → 1n−1↓ ↓ ↓0 → H 2 (G) → ˜K(G, n) → K(G, n) → 1where the maps downwards are isomorphisms.PROOF. Let F d /N be a presentation **of** G. Since [K(F d , n), S n ] equals K(F d , n)the same is true for the quotient by the affine subgroup R. Hence we have[X, S n ] = X. Also, identifying [K(F d , n), S (1)n−1] with K(F d , n − 1) we concludethat [X, S (1)n−1 ] is the same as ˜K(G, n − 1). Using the exact sequence **of** thestatement **of** Theorem 5.10 we concludeY def= X Sn /X S(1)n−1Applying Proposition 3.9 we get our statement.↓= K(G, n)/K(G, n − 1) ∼ = G.Corollary 5.14 Let n ≥ 3 and G be a finitely generated group. We choose apresentation F d /N ∼ = G **of** G. Then there exists a short exact sequence1 → [F d , F d ]/[F d , N] → ˜K(G, n) → G n−1 → 1.If G is perfect then the group on the left is just its universal central extension.55□

- 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 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 and 62: First, assume that i = n. ThenNow a
- Page 63 and 64: We note that for affine subgroups n
- Page 65: Every element of H 2 (G) maps to an
- 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
- Page 87 and 88: Remark 6.8 Proposition 6.5 shows us
- Page 89 and 90: 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
- 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.