The Space of Sections as a Topological Space
In this chapter we shall shift our focus from an individual section to the space of all sections at once. The space of sections naturally forms a pro-discrete topological space, see Lemma 44, which allows important limit arguments in arithmetically releva
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The Space of Sections as a Topological Space
In this chapter we shall shift our focus from an individual section to the space of all sections at once. The space of sections naturally forms a pro-discrete topological space, see Lemma 44, which allows important limit arguments in arithmetically relevant cases, see Lemma 48. The fundamental notion of a neighbourhood of a section is introduced and used to describe the decomposition tower of a section.
4.1 Sections and Closed Subgroups Let be a profinite group. The set Sub./ of closed subgroups of allows a description as a profinite set
Sub./ ! lim Sub.G/ G
via the projection to the finite quotients G of . We consider Sub./ as a topological space with respect to this profinite topology. For a short exact sequence 1 ! ! ! ! 1 of profinite groups the set Sub./= of -conjugacy classes of closed subgroups of allows similarly a description as a profinite space Sub./= ! lim Sub.G/='./: 'WG
The space of sections. The set of conjugacy classes of sections S! is naturally a subspace S! Sub./=
J. Stix, Rational Points and Arithmetic of Fundamental Groups, Lecture Notes in Mathematics 2054, DOI 10.1007/978-3-642-30674-7 4, © Springer-Verlag Berlin Heidelberg 2013
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4 The Space of Sections as a Topological Space
by assigning to a class of sections Œs the conjugacy class of images s. / . We endow S! with the induced topology. Unfortunately, the subspace S! is not necessarily closed, in other words compact, as the following lemma illustrates. Lemma 43. If is finite, then S! is a discrete topological space. Proof. Let s be a section of ! . The kernel 0 of the -action on by conjugation via s yields a normal subgroup s.0 / of . The corresponding quotient G D =s.0 / sits in a short exact sequence of finite groups 1 ! ! G ! =0 ! 1: The only class Œt 2 S! with a representative t such that t. / maps to the subgroup s. /=s.0 / G is the class of s.
t u
Characteristic quotients. A topologically finitely generated profinite group is in several natural ways a projective limit of characteristic finite quotients along the index system .N;
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