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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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