Local inverses of Borel homomorphisms and analytic P-ideals

Sławomir Solecki

doi:10.1155/aaa.2005.207

Abstract

We present a theorem on the existence of local continuous homomorphic inverses of surjective Borel homomorphisms with countable kernels from Borel groups onto Polish groups. We also associate in a canonical way subgroups ofℝwith certain analytic P-ideals of subsets ofℕ. These groups, with appropriate topologies, provide examples of Polish, nonlocally compact, totally disconnected groups for which global continuous homomorphic inverses exist in the situation described above. The method of producing these groups generalizes constructions of Stevens and Hjorth and, just as those constructions, yields examples of Polish groups which are totally disconnected and yet are generated by each neighborhood of the identity.

Highlights

Local homomorphisms are of importance in the study of Lie groups [1, 8]. They will appear in the more general context of Polish groups. (For a definition of Polish groups and other topological notions, see the end of the introduction.) We say that a homomorphism π : H → G between two topological groups is locally invertible by a local homomorphism if there exist a symmetric neighborhood U of 1 in G and a local homomorphism f : U → H such that π( f (g)) = g for all g ∈ U
Assume that each nonempty open subset of D generates D and that each continuous local homomorphism of D into a metric Borel group extends to a homomorphism defined on D
We first prove that a continuous local homomorphism f : U → B defined on an open symmetric neighborhood U of 1 in G can be extended to G

Summary

Introduction

The example produced by Stevens in [10], in answer to a question from the Scottish Book, was the first group which was Polish and both totally disconnected and generated by any neighborhood of 1. Another example of this sort was constructed by Hjorth [6]. A metric separable space is zero dimensional if it has a topological basis consisting of closed-and-open sets. Let Q2 stand for the group of diadic rationals, that is, all rational numbers whose denominators are powers of 2 with addition as the group operation

Local inverses

Subgroups of R and analytic P-ideals