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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

If we fix an isomorphism

If we fix an isomorphism between the G-fundamental groups of X with respect tox 0 and x 1 there is the following compositionI x1 ↩→ π top1 (X, G, x 1 ) ∼ = πtop1 (X, G, x 0 ) ↠ G.Even though the isomorphism in the middle is only well-defined up to conjugationby an element of π top1 (X, x 0 ) the whole composition always coincides with theinclusion map of I x1 into G.In terms of automorphisms of the universal G-cover ˜p G : ˜X × G → X wefix a point ˜x 1 on the fibre ˜p −1 G (x 1 ). Given an element g of I x1 there is a uniqueautomorphism ϕ g of ˜X × G that sends ˜x 1 to g · ˜x 1 . However, this automorphismreally depends on the choice of ˜x 1 .We now let R be a path connected subset of X that contains the point x 1 . Ifwe forget the G-action for a moment then ˜p −1 G (R) is a disconnected topologicalcover of R if G is non-trivial. We let ˜R be a component of ˜p −1 G (R) on ˜X × {1}.The group I R acts on ˜R simply by interchanging the |I R | different but homeomorphiccomponents. We choose a point of ˜R above x 1 to obtain an isomorphismof π top1 (X, G, x 0 ) with Aut(˜X × G). In this special situation we see that an automorphismcorresponding to an element of I R depends only on ˜R and not on theparticular point lying above x 1 . Hence it makes sense to talk about an automorphismof the universal G-cover that is the inertia automorphism of a componentof ˜p −1 G (R).We finally want to stress that in general there is no natural way of relatingelements of G to cover automorphisms of ˜X×G or elements of the G-fundamentalgroup of X since the G-action usually does not respect the fibres. It is only inertiathat makes this possible.Given a G-cover p : Y → X there is always an injection of inertia groupsI y ⊆ I p(y) for all points y ∈ Y. Given a cover Z → G\X we can form the fibreproduct with X and obtain a G-cover p ′ : Z × G\X X → X. All points z on thisfibre product fulfill I z = I p ′ (z). Conversely, if p : Y → X is a G-cover thatfulfills I y = I p(y) for all points y of Y then the quotient by G defines a coverG\p : G\Y → G\X. Hence there is a one-to-one correspondence{ }{ } G-covers p : Y → X such thatcovers of G\X ↔I y = I p(y) for all y ∈ YSince this remains true if we assume connectivity on both sides of this correspondencethe induced homomorphism of fundamental groupsπ top1 (X, G, x 0 ) ↠ π top1 (G\X, ¯x 0 )is surjective. Here, ¯x 0 denotes the image of x 0 under the quotient map X → G\X.36

For an element g of the inertia group I x1 we denote by ı g the image of g inπ top1 (X, G, x 1 ) as constructed above. We let p : Y → X be a G-cover withI y = I p(y) for all points of Y. If we fix a point ˜x on ˜X then there is a uniquecover automorphism ϕ g of ˜X × G such that ϕ g · ˜x = ˜x · ı g . By our assumptionson the inertia groups of Y this automorphism ϕ g will act trivially on Y. So thesubgroup N normally generated by all inertia elements lies in the kernel of thehomomorphism from π top1 (X, G, x 0 ) onto π top1 (G\X, ¯x 0 ). Conversely, the quotientof the universal G-cover by N is a G-cover q : Z → X with I z = I q(z) for all pointsz of Z. Hence π top1 (X, G, x 0 )/N is a quotient of π top1 (G\X, ¯x 0 ). But this meansthat N is precisely the kernel we are looking for. So we obtain a short exactsequence1 → N → π top1 (X, G, x 0 ) → π top1 (G\X, ¯x 0 ) → 1.As a special case we obtain the following: If G acts without fixed points on X thenX → G\X is a regular cover with group G, there are no non-trivial inertia groupsand we just get the well-known short exact sequence1 → π top1 (X, x 0 ) → π top1 (G\X, ¯x 0 ) → G → 1.4.4 Loops and the orbifold fundamental groupThe material of this section should be well-known. However, the author could notfind a reference for it.As in the previous section we let X be normal irreducible complex analyticspace and G be a finite group of automorphisms of X. We keep all notationsintroduced so far.We will always assume that the quotient space G\X is smooth, i.e. a complexmanifold. By purity of the branch locus the branch locus D of q : X → G\X is adivisor, cf. [GR1, Satz 4]. We denote by D i , i = 1, ..., r the irreducible componentsof this divisor, cf. [GR2, Chapter 9.2.2]. We denote by e i the ramificationindex of D i .The inertia groups of the components of q −1 (D i ) for fixed i are conjugatesubgroups of G. These components are divisors and so their inertia groups must becyclic. More precisely, every inertia group of a component of q −1 (D i ) is abstractlyisomorphic to the cyclic group ¡ e i.Given a G-cover p : Y → X we form the quotient G\p : G\Y → G\X.Outside ⋃ i D i this is a topological cover. This defines a homomorphism fromthe fundamental group of G\X − D to the G-fundamental group of X. If Y isconnected as a G-cover then its quotient is also connected. We assumed X tobe normal so also Y must be normal and so the same is true for the quotient37

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