2 years ago

On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

For both groups π top1

For both groups π top1 (¢ 2 − D) and π topPROOF.1 ( 2 − D) there are surjectivehomomorphisms ψ onto S n that are compatible with ı ∗ . Since δ is trivial inπ top1 ( 2 − D) we conclude that ψ(δ) = 1.By a theorem of Oka (cf. [FL, Corollary 8.4]) the short exact sequence0 → A → π top1 (¢ 2 − D)ı ∗→πtop1 ( 2 − D) → 1is central. We know that A is normally generated by δ and hence δ must be acentral element of π top (¢ 1 2 − D). Of course δ remains central in every quotient ofπ top (¢ 1 2 − D).We recall the short exact sequences1 → π top1 (X affgal ) → πtop 1 (¢ 2 − D)/ ≪ Γ i 2 ≫ ψ → S n → 1↓ ↓ ı ∗ ||1 → π top1 (X gal ) → π top1 ( 2 − D)/ ≪ Γ i 2 ≫ ψ → S n → 1We already noted that the kernel of the surjective homomorphism ı ∗ is generatedby δ. Since ψ(δ) = 1 the loop δ lies in π top1 (Xgal aff ). This yields the first exactsequence.There exist surjective homomorphismsπ top1 (Xgal aff,n−1 ) ↠ πtop 1 (X aff )↓↓π top1 (X gal , S (i)n−1) ↠ π top1 (X)The kernel of the upper horizontal homomorphism N is generated by inertiagroups. The kernel of the lower horizontal homomorphism is generated by theimage of N from above. The kernel of the left arrow downwards is generated byδ. Chasing around this diagram we find that the kernel of the surjective map fromπ top1 (X aff ) onto π top1 (X) is generated by ¯δ. □In Theorem 6.2 we constructed an isomorphismπ top1 (X affgal )/Caff ∼ =˜K(πtop1 (X aff ), n).Since δ is central it is stable under the S n -action on the right. The same holds truewhen passing to the quotient K(π top1 (X aff ), n). So if we consider K(π top1 (X aff ), n)as a subgroup of π top1 (X aff ) n then δ maps to an element of the diagonal. AndProposition 6.4 tells us exactly what this element is:72

Proposition 6.5 Under the isomorphism of Theorem 6.2 and the surjective mapof Theorem 4.7 the loop δ maps as followsπ top1 (Xgal aff)/Caff∼ top = ˜K(π 1 (Xgal aff ), n) ↠ K(πtop 1 (X aff ), n)δ ↦→ δ ↦→ (¯δ, ..., ¯δ)where ¯δ is the central element of Proposition 6.4.□Again, we can say a little bit more for the abelianisationCorollary 6.6 We keep the notations and assumptions of Theorem 6.2. Then wedenote by ¯C proj the image of C proj in the abelianised fundamental group of X gal .Then there exists an isomorphismH 1 (X gal , ¡ )/ ¯C proj ∼ = H1 (X, ¡ ) ⊕ H 1 (X aff , ¡ ) n−2 .In particular, if π top1 (X aff ) is abelian thenπ top1 (X gal )/C proj ∼ = πtop1 (X) × π top1 (X aff ) n−2 .We note that these isomorphisms are not canonical.To increase readability, we abbreviate H 1 ¡ (−, ) just by H 1 (−).Since abelianisation is not an exact functor we have to proceed by hand:PROOF.1 → 〈δ〉 → π top1 (Xgal aff)→ πtop 1 (X gal ) → 1↓ ↓ ↓〈δ ′ 〉 ↩→ H 1 (X affgal ) ↠ H 1(X gal )where δ ′ denotes the image of δ in H 1 (Xgal aff).Let x be an element of H 1(X affthat maps to 0 in H 1 (X gal ). We can lift this to an element ˜x of π top1 (Xgal aff)thathas to map to a product of commutators in π top1 (X gal ) by commutativity of thediagram. But this means that ˜x is a product of δ s for some integer s times somecommutators. Changing ˜x by commutators we still get a lift of x. So we mayassume that ˜x actually equals δ s . Therefore, x is equal to ¯δ s . This shows that wehave an exact sequence1 → 〈¯δ〉 → H 1 (X affgal ) → H 1(X gal ) → 1.We denote by ¯δ the image of δ in π top1 (X aff ) we know from Proposition 6.4 that thesubgroup generated by ¯δ inside π top1 (X aff ) is equal to the kernel of the projectionπ top1 (X aff )↠π top1 (X). So we obtain another exact sequence1 → 〈¯δ ′ 〉 → H 1 (X aff ) → H 1 (X) → 173gal )

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