MSI Weekly Bulletin - Week starting Monday 6 October, 2008

Spaltenstein, and later Böcksted and Neeman explained how to construct injective resolutions for unbounded complexes, but very little is known about "relative" homological algebra in this setting (that is we wish to change the class of injectives). I will present an approach developed with Chacholski and Pitsch, and mostly concentrate on the case when the new class of injectives is contained in the class of usual injectives.

The Riemann Mapping Theorem is equivalent to mapping a planar domain to the unit disk by a quasiconformal (QC) mapping. These QC mappings generalize to higher dimensions (where conformal mappings are trivial). The main problem was to characterize the QC images of the unit ball, i.e., QC balls. In his reviews Ahlfors emphasized the related question of characterizing the images of the unit sphere under QC mappings of space, i.e., quasispheres. In their plenary talks at the ICM, both Ahlfors and Gehring reiterated these questions. Our solution comes from reflection: a sense-reversing idempotent homeomorphism of R3 onto itself. Any reflection F of the sphere