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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

Given an arbitrary

Given an arbitrary connected G-cover Z → X we can form its fibre productwith X × G. Then we can find a connected component Z ′ (with respect to C G ) ofthis fibre product that dominates Z. Since Z ′ is a G-cover of X × G it is dominatedby ˜X × G. From this we conclude that ˜X × G is a universal cover of C G .We define ˜p G : ˜X × G → X with the G-action as above and denote its groupof G-automorphisms by Aut(˜X × G). For every point x 0 of X, this latter groupacts from the left on the fibre ˜p G −1 (x 0 ).Definition 4.4 For a point x 0 of X and a group G that acts by automorphisms onX we denote the opposite group of Aut(X × G) by π top1 (X, G, x 0 ) and call it theG-fundamental group of X.As in the case of the classical fundamental group to give a G-cover is the same asto give a discrete set with a right action of π top1 (X, G, x 0 ) on it.Given a subgroup H of G the fundamental groups classifying covers withactions of H and G are related as follows: We fix a system R of representativesof G/H. We will assume that the class of H is represented by the unit element ofG. For a connected C H -cover p : Y → X we define the following C G -cover:and a left G-action on Y × R viaY × R → X(y, r) ↦→ r · p(y)G × (Y × R) → Y × Rg , (y, r) ↦→ (h g y, r g h)where g = r g h g is the unique decomposition of an element of G into a productof an element of H and an element of R. This clearly is a connected object ofC G . The object so associated to Y is the same as the fibre product of Y withq : X × R → X with the G-action described above. This is an exact functor fromC H to C G and hence defines an injective homomorphism of fundamental groupsπ top1 (X, H, x 0 ) ↩→ π top1 (X, G, x 0 ).With respect to the action of π top1 (X, G, x 0 ) on the fibre q −1 (x 0 ) the image of thishomomorphism is the stabiliser of the point (x 0 , 1) of X × R.Given an element γ of π top1 (X, G, x 0 ) it acts on the fibre q −1 (x 0 ) of the G-cover q : X × R → X by sending (h −1 x 0 , h) to (h −1 r(γ) −1 x 0 , r(γ)h). Moreover,if H is a normal subgroup of G then the map that sends γ to r(γ) defines a homomorphismfrom π top1 (X, G, x 0 ) to G/H. This homomorphism is surjective sincewe can lift the map x ↦→ g·x to the universal cover as explained in [Di, Satz I.8.9].Hence there exists a short exact sequence1 → π top1 (X, H, x 0 ) → π top1 (X, G, x 0 ) → G/H → 1.34

In particular, for H = 1 we obtain the short exact sequence1 → π top1 (X, x 0 ) → π top1 (X, G, x 0 ) → G → 1. (∗)To obtain an isomorphism of π top1 (X, G, x 0 ) with Aut(˜X × G) we have to choosea point on the fibre ˜p −1 G (x 0 ).If we choose another base point, say x 1 on X then the G-fundamental groupswith respect to two x i ’s are isomorphic. However, such an isomorphism dependson the choice of points ˜x i in the respective fibres ˜p −1 G (x i ), i = 0, 1. We willassume that the two ˜x i ’s lie on the same topological component of the universalG-cover ˜X × G. This means that we choose a path connecting x 0 to x 1 . Then theisomorphismπ top1 (X, G, x 0 ) ∼ = πtop1 (X, G, x 1 )is well-defined up to conjugation by an element of π top1 (X, x 0 ) and the two homomorphismsonto G coming from the short exact sequence (∗) are compatibleunder this isomorphism.For a G-cover p : Y → X and a closed subset of A of Y we call the subgroupof G fixing A pointwise the inertia group of A (in G):I A := {g ∈ G | ga = a, ∀a ∈ A}.The possibly larger subgroup of G fixing A but not necessarily pointwise is calledthe decomposition group of A (in G):D A := {g ∈ G | g(A) = A}.The inertia group is always a normal subgroup of the decomposition group.We choose a point x 1 on X and let ˜p G : ˜X × G → X be the universal G-cover. Then the inertia group I x1 acts on the fibre ˜p −1 G (x 1 ). We choose a point˜x 1 on this fibre. Then we compare the left action of I x1 with the right action ofπ top1 (X, G, x 1 ) in this point ˜x 1 . This associates to each element of I x1 an elementof π top1 (X, G, x 1 ). Given another point ˜x ′ 1 above x 1 there is a G-automorphism ϕthat sends ˜x 1 to ˜x ′ 1. We assume that g · ˜x 1 = ˜x 1 · γ g for an element g of I x1 . Sinceϕ is G-equivariant we computeg · ˜x ′ 1 = g · (ϕ(˜x 1)) = ϕ(g · ˜x 1 ) = ϕ(˜x 1 · γ g ) = ϕ(˜x 1 ) · γ g = ˜x ′ 1 · γ g.Hence γ g does not depend on the choice of a point in the fibre above x 1 and itacts like multiplication by g on all points on this fibre. This means that there is anatural injective homomorphismI x1 ↩→ π top1 (X, G, x 1 ).35

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