Any group arising as the profinite completion of some group is a profinite group. We treat g k as a fundamental object of study because it allows us to control all separable extensions l of k in one stroke. Such profinite gspaces occur naturally when we look at the etale topological type of a scheme x defined over a field k. Profinite completion and doubledual internet archive.
A topology on a set x is a collection t of subsets of x, satisfying the following axioms. In particular, we generalize artinmazurs comparison theorem 2, 12. Browse other questions tagged abstractalgebra generaltopology grouptheory topologicalgroups profinitegroups or ask your own question. Geom dedicata profinite rigidity for seifert fibre spaces gareth wilkes 0 0 mathematical institute, university of oxford, andrew wiles building, radcliffe observatory quarter, woodstock road, oxford ox2 6gg, uk an interesting question is whether two 3manifolds can be distinguished by computing and comparing their collections of finite covers. By stone duality these are equivalent to stone spaces and thus are often called profinite spaces. A profinite space group is the projective limit of finite sets groups. This is further complicated by the question of whether or. This applies, for example, to the definitions of interior, closure, and frontier in pseudometric spaces, so these definitions can also be carried over verbatim to a topological space. For a pathconnected topological space x, the fundamental group.
So these are compact hausdorff totally disconnected topological spaces. Suppose that fis continuous and let a y be a closed set. Profinite graphs and groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. This is not the most convenient characterization of a profinite space. Cofinite graphs and their profinite completions acharyya. Whenever we are given two topological groups, we insist that a homomorphism between them be continuous. Let us recall from 14 pro nite spaces and pro nite completions.
For instance, wilton and zalesskii have shown that the geometry if any of a 3manifold is detected by the profinite completion. The padic topology is the emblematic example of a pro. In this context, an inverse system consists of a directed set. But, to quote a slogan from a tshirt worn by one of my students. A group is a residually finite group if and only if the natural map to its profinite completion is injective. K is closed in k for every compact subspace k of x. For any ring r, direct limits exist in the category rmod of rmodules. A topological group is a group g which is also a topological space with the property that the multiplication map m. Given this definition, it is possible to recover the first definition using the inverse limit lim profinite topological spaces 1843 2. The topological semigroups thus obtained are socalled pro c semigroups. Most importantly, the notion of profinite completion becomes completely transparent in this set up, and it is leftadjoint to the canonical inclusion of profinite spaces into prospaces, and hence. In johnstones stone spaces it is proved that the category of profinite partial orders is equivalent to the category of ordered stone spaces also called priestley spaces and that the obvious. A random walk on the profinite completion of z sciencedirect. Continuous cohomology and homology of profinite groups.
The survey includes several concrete applications to automata theory. Profinite completions and canonical extensions of heyting. Let obc be the collection of topological spaces, with morphisms being continuous maps. We will soon prove that every metric space has a completion. The main purpose of part i of these notes is to develop for a ring r a functional notion of rcompletion of a space x. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further. Suppose h is a subset of x such that f h is closed where h denotes the closure of h. A topological group is a set gwith both the structure of a group, and of a topological space, such that the multiplication law g g. Direct limits, inverse limits, and profinite groups math 519 the rst three sections of these notes are compiled from l, sections i. One of the key technical ingredients for the comparison theorem for algebraic stacks via topological types is the fact that the pro nite completion on prosimplicial sets admits. The completion of a topological group in hartleys work can be thought of as a particular example of the completion of a general uniform space. In particular, direct limits exist in the category ab of abelian groups. Profinite rigidity for seifert fibre spaces pdf paperity. For the class c of all finite semigroups, the attribute pro c becomes simply profinite.
The little 2disk operad is an operad in topological spaces. Introduction to topological groups dikran dikranjan to the memory of ivan prodanov abstract these notes provide a brief introduction to topological groups with a special emphasis on pontryaginvan kampens duality theorem for locally compact abelian groups. For rzp and x subject to usual finiteness condition, the rcompletion coincides up to homotopy, with the pprofinite completion of quillen and sullivan. In this section, we study profinite completion of topological types. The category eb of compact, hausdor, and totally disconnected topological spaces is equivalent to the pro. In the dictionary between number fields and function fields, one thinks as galois groups as followstake the etale fundamental group of your curve minus some points, and take the limit as one removes all the points. Y between topological spaces is called continuous if f 1u is open in xfor each set uwhich is open in y.
It has as a basis of open subsets all left cosets of subgroups of finite index it has as a basis of open subsets all right cosets of subgroups of finite index it has as a basis of open subsets all cosets of normal subgroups of finite index. In particular, seifert fibre spaces are distinguished. We follow for the profinite completion of prosimplicial sets. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. Articles in pdf format may be submitted by email directly to a transmitting editor. Then, as a special situation, we study cofinite graphs and their uniform completions. The pro nite completion of the integers, the padic integers. In mathematics, a profinite integer is an element of the ring topological space is profinite if it is homeomorphic to a limit of a diagram of finite discrete spaces. Most importantly, the notion of profinite completion becomes completely transparent in this set up, and it is leftadjoint to the canonical inclusion of profinite spaces into pro spaces, and hence. Katonakayama spaces, infinite root stacks, and the profinite. The entry linked to here treats another more purely topological concept. Metricandtopologicalspaces university of cambridge. A basis of neighbourhoods of the identityelement consists of all normal subgroups of nite index. It was introduced by may and boardmanvogt in order to describe the structure existing on the 2fold loops on a simply connected based space that allows one to recover that space up to weak equivalence see may72 for details about this theorem.
Preliminaries in this section we recall the basic topological notions that are used in the paper. Nov 29, 2016 these examples notwithstanding, the profinite completion of the fundamental group of a lowdimensional manifold is known to contain a large amount of information. A profinite group is a hausdorff, compact, and totally disconnected topological group. Paper 1, section ii 12e metric and topological spaces. Continuous group actions on profinite spaces request pdf. We characterize dual spaces of canonical extensions of bounded distributive lattices and heyting algebras in terms of nachbin ordercompactifications.
Topological space definition of topological space by. One checks that a discrete topological space is totally disconnected, and that a product of totally disconnected spaces is totally disconnected, and that being totally disconnected is hereditary. The profinite completion of is the inverse limit of the inverse system of all finite quotients of, with maps as follows. Similarly f injective for all finite pgroups if jp is an epimorphism. Denote by bh t the pro nite completion of the topological type ht which is a priori a prosimplicial set. For a log scheme locally of finite type over c, a natural candidate for its profinite homotopy type is the profinite completion of its katonakayama space. Since ynais open, f 1yna is open and therefore f 1a xnf 1yna is closed. A similar discussion applies in the topological case. This is provided by the fibrant replacement functor in. The homotopy category of topological spaces has a good combina torial model provided by simplicial sets. The profinite topology on a group is a topology on the underlying set of the group defined in the following equivalent ways. Y between topological spaces is continuous if and only if the inverse image of every closed set is closed. In the context of topological groups, we also require continuity for homomorphisms.
In mathematics, a profinite integer is an element of the ring 0 such that the open ball b y u and x 2 b 2 y. In the dictionary between number fields and function fields, one thinks as galois groups as followstake the etale fundamental group of your curve minus some points, and take the limit as one removes all the. A free pro nite group is the pro nite completion of a free group. We generalize the idea of cofinite groups, due to b. Beware there are two possible interpretations of this term. Topological space definition is a set with a collection of subsets satisfying the conditions that both the empty set and the set itself belong to the collection, the union of any number of the subsets is also an element of the collection, and the intersection of any finite number of the subsets is an element of the collection. In mathematics, padic analysis is a powerful tool of number theory. Zare continuous functions between topological spaces.
We show that the profinite completions and canonical extensions of bounded distributive lattices and of boolean algebras coincide. A random walk on the profinite completion of a finitely. It has as a basis of open subsets all left cosets of subgroups of finite index. A topological space x is called compactly generated if it satis. One is handled in the section above, being profinite completion of the homotopy type of a space. Profinite groups can be defined in either of two equivalent ways. Just as the term space is used by some schools of algebraic topologists as a synonym for simplicial set, so profinite space is sometimes used as meaning a simplicial object in the category of compact and totally disconnected topological spaces, i. Profinite rigidity for seifert fibre spaces springerlink. A profinite group which is not its own profinite completion. Topological types of algebraic stacks international. Youre probably aware of this, but heres an analogy which might explain why both of these observations are to be expected. These examples notwithstanding, the profinite completion of the fundamental group of a lowdimensional manifold is known to contain a large amount of information. T2 the intersection of any two sets from t is again in t. A topological space is profinite if it is homeomorphic to the inverse limit of an inverse.