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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

If τ 1 and τ 2 have

If τ 1 and τ 2 have exactly one index in common then there is a point with inertiagroup S 3 that is generated by r 1 and r 2 . So there is a triple commutator relationbetween r 1 and r 2 and so also c(r 1 , r 2 ) = 1 holds true in this case.So if Question 2.14 has an affirmative answer for the universal cover ˜X gal ofX gal then all the c(r 1 , r 2 )’s are equal to 1 and so C proj is trivial.□Again, everything said so far can also be done in the affine setup. We thendefine C aff to be the normal subgroup of π top1 (Xgal aff,S n) defined by the c(r 1 , r 2 )’swhere the r i ’s run through the inertia groups corresponding to some universalS n -cover of Xgal aff . We then getTheorem 4.7 Let f : X →2 be a good generic projection of degree n withGalois closure X gal . Then there are surjective homomorphismsπ top1 (X gal ) ↠ π top1 (X gal )/C proj ↠ K(π top1 (X), n)π top1 (Xgal aff)↠ πtop 1 (Xgal aff)/Caff↠ K(π top1 (X aff ), n).If Question 2.14 has an affirmative answer for the universal cover of X affgal thenboth C aff and C proj are trivial.If Question 2.14 has an affirmative answer for the universal cover of X gal thenat least C proj is trivial.Again, even if C aff is trivial we cannot expect these surjective homomorphisms tobe isomorphisms. We refer to Theorem 6.2 for details.Corollary 4.8 For a good generic projection f : X →surjective and non-canonical homomorphisms2 of degree n there areH 1 (X gal , ¡ ) ↠ (H 1 (X, ¡ )) n−1H 1 (X affgal , ¡ ) ↠ ( H 1 (X aff , ¡ ) ) n−1 .PROOF. From Morse theory ([Mil]) it is known that smooth affine and smoothprojective varieties are CW-complexes. So we can apply Hurewicz’s theorem thatH 1 ¡ (−, ) is isomorphic to the abelianised fundamental group.Thus our statement follows from the fact that K(−, n) for n ≥ 3 commuteswith abelianisation by Proposition 3.8 and the computation of K(−, n) for abeliangroups given by Corollary 3.5.□42

5 A generalised symmetric group5.1 Definition of S n (d)Seht doch hinab! Im Mondschein auf den GräbernHockt eine wild-gespenstische Gestalt!Ein Aff ist’s! Hört ihr, wie sein HeulenHinausgellt in den süßen Duft des Lebens?We let τ k be the transposition (k k + 1) of S n . From the theory of Coxeter groups(cf. also Section 5.6) it is known that S n admits a presentation asS n = 〈 τ k , k = 1, ..., n − 1 | τ k 2 , (τ k τ k+1 ) 3 , (τ k τ j ) 2 ∀|k − j| ≥ 2 〉 .Let d ≥ 1 and n ≥ 3 be natural numbers. We want to construct a generalisedsymmetric group where we have d copies of the transposition (1 2). For this welet s 1 , ..., s d be free generators of the free group F d of rank d. Then we define thegroup( )˜S n (d) := F d ∗ S (1)n−1 /Rwhere R is the subgroup normally generated by the following elementss i2for i = 1, ..., d(s i · τ 2 ) 3 for i = 1, ..., d(s i · τ k ) 2 for k ≥ 3 and i = 1, ..., d.The reader will identify this group as the d-fold amalgamated sum of S n withitself where we amalgamate the subgroup S (1)n−1 in every summand.Every summand has a map (the identity) onto S n that is compatible with thesubgroup that is amalgamated. These homomorphism patch together to a homomorphismψ onto S n . Sending S n via the identity to the first summand we obtaina splitting ϕ of ψ.But we still want more relations to hold true: We defineS n (d) := ˜S n (d)/R ′where R ′ is the subgroup normally generated by the following elements:(ϕ(σ)s i ϕ(σ) −1 · s j ) 2(ϕ(σ)s i ϕ(σ) −1 · s j ) 3if σ(1 2)σ −1 and (1 2) are nodal transpositionsif σ(1 2)σ −1 and (1 2) are cuspidal transpositionsThe homomorphisms ψ and ϕ induce homomorphisms on the quotient S n (d) thatwe will call by abuse of notation again by ψ and ϕ.43

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