A decomposition theorem for compact groups with application to supercompactness

TL;DR

This paper proves that every compact connected group can be approximated by inverse sequences with specific bonding maps, leading to a new proof that all compact groups are supercompact, extending the understanding of their structure.

Contribution

It introduces a decomposition theorem for compact groups and applies it to establish that all compact groups are supercompact, providing a new proof of a longstanding result.

Findings

Every compact connected group is the limit of a specific inverse sequence.

All compact groups are supercompact, as shown by the new proof.

The decomposition involves epimorphisms with finite kernels and projections from products with simple Lie groups.

Abstract

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.