nLab
Bohr compactification

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

Bohr compactification is a kind of compactification of a topological group to one that is compact Hausdorff.

Definition and existence

Let i:CHGrpTopGrpi: CHGrp \to TopGrp denote the full embedding of the category of compact Hausdorff topological groups into the category of all topological groups. Let GG be a topological group.

Definition

The Bohr compactification Bohr(G)Bohr(G) is a representing object for the functor TopGrp(G,i):CHGrpSetTopGrp(G, i-): CHGrp \to Set.

Theorem

The Bohr compactification exists for any topological group GG.

Proof

We verify that the hypotheses of the general adjoint functor theorem are satisfied. Certainly CHGrpCHGrp is complete, and i:CHGrpTopGrpi: CHGrp \to TopGrp preserves small limits since products and equalizers in CHGrpCHGrp are formed just as they are in TopGrpTopGrp. We must now show there is a small solution set, i.e., a weakly initial set for the comma category GiG \downarrow i.

Let I=[0,1]I = [0, 1] be the unit interval with its standard topology. There is a canonical map u G:GI Top(G,I)u_G: G \to I^{Top(G, I)} where u:idMu: id \to M is the unit of the monad M=I Top(,I)M = I^{Top(-, I)} on TopTop; note u Xu_X is a closed embedding if XX is compact Hausdorff (see the discussion here). We claim that the collection of closed subspaces of I Top(G,I)I^{Top(G, I)} that come equipped with topological group structures is a solution set. Indeed, for an arbitrary CH group KK and continuous homomorphism f:GKf: G \to K, the closure of the image f(G)¯\widebar{f(G)} is a compact Hausdorff subgroup of KK through which ff factors (according to the lemma that follows), and this is a closed subspace of M(f) 1(u K)M(f)^{-1}(u_K) in the pullback diagram

M(f) 1(u K) I Top(G,I) M(f) K u K I Top(K,I)\array{ M(f)^{-1}(u_K) & \hookrightarrow & I^{Top(G, I)} \\ \downarrow & & \downarrow_{M(f)} \\ K & \underset{u_K}{\hookrightarrow} & I^{Top(K, I)} }

and therefore f(G)¯\widebar{f(G)} occurs as a closed subspace of I Top(G,I)I^{Top(G, I)}, as claimed.

Lemma

If HH is a topological group and JHJ \subseteq H is a subgroup, then the topological closure J¯H\widebar{J} \subseteq H is also a subgroup. (The same is true for any finitary algebraic theory in place of the theory of groups.)

Proof

For X,YX, Y arbitrary topological spaces and subsets AXA \subseteq X, BYB \subseteq Y, it is elementary that A×B¯=A¯×B¯\widebar{A \times B} = \widebar{A} \times \widebar{B}. Hence for any operation m:H nHm: H^n \to H that JJ is closed under, J¯ n=J n¯\widebar{J}^n = \widebar{J^n} is contained in m 1(J¯)m^{-1}(\widebar{J}) (by continuity of mm), so that mm restricts to an operation J¯ nJ¯\widebar{J}^n \to \widebar{J} as required.

Remark

The Bohr compactification admits a more elegant construction if GG is a topological abelian group: if S=S 1S = S^1 is the unit circle, then Bohr(G)Bohr(G) may be taken to be the closure of the image of GG under the canonical map GS TopAb(G,S)G \to S^{TopAb(G, S)}. The argument is that it suffices to consider only compact Hausdorff abelian groups KK, where KK embeds as a closed subgroup of S TopAb(K,S)S^{TopAb(K, S)} by Pontryagin duality; using a pullback similar to the above, the argument is easily completed. The description simplifies further if GG is a locally compact Hausdorff abelian group; here Bohr(G)Bohr(G) is the Pontryagin dual of the discretization of the Pontryagin dual of GG.

Example

A MathOverflow question from 2011 asks whether, for GG a compact Hausdorff group, can \mathbb{Z} appear as a quotient of GG considered as an abstract group?

A truly simple answer was given by Sean Eberhard using the Bohr compactification, as follows. A quotient p:Gp: G \to \mathbb{Z} admits a section i:Gi: \mathbb{Z} \to G which extends to the Bohr compactification i^:Bohr()G\widehat{i}: Bohr(\mathbb{Z}) \to G. The composite pi^:Bohr()p \widehat{i}: Bohr(\mathbb{Z}) \to \mathbb{Z} is still surjective as its restriction along the unit Bohr()\mathbb{Z} \to Bohr(\mathbb{Z}) is the identity. According to the remark above, Bohr()Bohr(\mathbb{Z}) is the Pontryagin dual of the discrete group S 1 disc/S^1 \cong \mathbb{R}_{disc} \oplus \mathbb{Q}/\mathbb{Z}, so Bohr() disc(/)Bohr(\mathbb{Z}) \cong \mathbb{R}_{disc}' \oplus (\mathbb{Q}/\mathbb{Z})'. Now the Pontryagin dual of a discrete torsionfree group such as disc\mathbb{R}_{disc} is divisible (more generally, if QQ is an injective module over a commutative ring and FF is flat, then hom(F,Q)\hom(F, Q) is also injective), so the restriction of pi^:Bohr()p \widehat{i}: Bohr(\mathbb{Z}) \to \mathbb{Z} to the summand disc\mathbb{R}_{disc}' is zero. But the restriction to the other summand (/)^= p p (\mathbb{Q}/\mathbb{Z})' \cong \widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p^\wedge is also zero, as the further restriction to 2 \mathbb{Z}_2^\wedge is zero by 3-divisibility, and the restriction to p2 p \prod_{p \neq 2} \mathbb{Z}_p^\wedge is zero by 2-divisibility.

References

Last revised on August 23, 2018 at 01:24:48. See the history of this page for a list of all contributions to it.