Corona Rigidity

TL;DR

This paper surveys how additional set theoretic axioms influence the rigidity of quotient structures, especially automorphisms, across Boolean algebras, ech-Stone remainders, and 4-algebras, highlighting key results and open problems.

Contribution

It provides a comprehensive overview of the effects of set theoretic axioms on the automorphism rigidity of quotient structures, including generalizations and future research directions.

Findings

Forcing axioms imply trivial automorphisms of 4() Fin quotients.

Under CH, non-trivial automorphisms of 4() Fin exist.

Survey covers Boolean algebras, ech-Stone remainders, and 4-algebras.

Abstract

We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal{P}(\mathbb{N})/\text{Fin}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of $\mathbb{N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal{P}(\mathbb{N})/\text{Fin}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, \v{C}ech-Stone remainders, and $\mathrm{C}^\ast$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.