The weak Extension Principle

TL;DR

The paper investigates rigidity phenomena for maps between Čech–Stone remainders $X^*$ under forcing axioms, focusing on the weak Extension Principle $\mathsf{wEP}(X,Y)$. It extends Farah's dimension-based reductions to locally compact noncompact second countable spaces and develops a lift-based framework via Gel'fand duality for abelian corona $\mathrm{C^*}$-algebras under $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$. The main result shows $\mathsf{wEP}$ holds unconditionally in the considered setting, implying no continuous surjection $(X^*)^\kappa\to (Y^*)^\lambda$ exists when $\kappa<\lambda$, while CH would allow many surjections; the proof assembles local liftings into a global one using a strong rigidity lifting theorem. This work advances the understanding of rigidity of maps between Čech–Stone remainders and informs the noncommutative extensions via corona algebras under set-theoretic axioms.

Abstract

We prove a rigidity result for maps between Čech-Stone remainders under fairly mild forcing axioms.

Theorems & Definitions (17)

• Definition 1.1
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Claim 2.4
• proof