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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

quotient does not depend

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 ofnotation we will also denote its restriction to K(F d , n) with p 1 . As a consequenceof 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□

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