Views
2 years ago

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

Groups of integral representation type - MSP
Carbon reductions generate positive ROI - Carbon Disclosure Project
The Path Ahead: Fundamentals of Generative ... - Right 4 Your Type
Renewable Projects Portal for CLP Group - OSIsoft
Generic process improvement approach applied to IT projects - PMI ...
Building Next Generation Design Support Programmes - See Project
Projected Costs of Generating Electricity - OECD Nuclear Energy ...
Project Insight - Power Generation Construction Projects in Asia-Pacific
GALOIS THEORY - Tata Institute of Fundamental Research
On The Structure of Certain Galois Cohomology Groups
The Fundamental Theorem of Galois Theory and Normal Subgroups
Generic Galois groups for q-difference equations
periods of eisenstein series: the galois case - Project Euclid
fields and galois theory - Neil Strickland - University of Sheffield
Introduction to Differential Galois Theory
Galois module structure of local unit groups
Chapter 4 COVERING PROJECTIONS AND FUNDAMENTAL GROUP
On Galois Algebras with an Inner Galois Group - Bradley Bradley
Supersingular Galois representations and a generalization of a ...
Galois groups with restricted ramification Romyar Sharifi Harvard ...
On Galois Algebras with a Unique Galois Group - Bradley Bradley
1 The Galois Group of a Quadratic 2 The Galois Group of a Cubic
A family of polynomials with Galois group $ PSL_5 (2) $ over ...
Automorphy lifting for residually reducible l-adic Galois ...
MILD PRO-p-GROUPS AND GALOIS GROUPS OF p-EXTENSIONS ...
Wildly ramified Galois representations and a generalization of a ...
ON GALOIS EXTENSIONS WITH AN INNER GALOIS GROUP ... - FUJI
The Frattini subgroup of the absolute Galois group - Tel Aviv University