Rigidity conjectures for continuous quotients

We prove several rigidity results for corona C*-algebras and Cech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todorcevc's OCA and Martin's Axiom at level (sic)(1) imply: (i) that if X and Y are locally compact second countable topological spaces, then all homeomorphisms between /3X \X and /3Y \ Y are induced by homeomor-phisms between cocompact subspaces of X and Y ; (ii) that all automorphisms of the corona algebra of a separable C*-algebra are trivial in a topological sense; (iii) that if A is a unital separable infinite -dimensional C*-algebra, the corona algebra of A circle times K(H) does not embed into the Calkin algebra. All these results do not hold under the Continuum Hypothesis.