In this, the second of a three-part series of talks, we describe a "definable Čech cohomology theory" strictly refining its classical counterpart. As applications, we show that, in strong contrast to its classical counterpart, this definable cohomology theory provides complete homotopy invariants for mapping telescopes of dd-tori and of dd-spheres; we also show that it provides an equivariant homotopy classification of maps from mapping telescopes of dd-tori to spheres, a problem raised in the d=1d=1 case by Borsuk and Eilenberg in 1936. These results build on those of the first talk. They entail, for example, an analysis of the phantom maps from a locally compact Polish space XX to a polyhedron PP; instrumental in that analysis is the definable lim1lim1 functor. They entail more generally an analysis of the homotopy relation on the space of maps from XX to PP, and we will begin by describing a category particularly germane for this analysis. Time permitting, we will conclude with some discussion and application of a related construction, namely that of the definable homotopy groups of a locally compact Polish space XX.
This is joint work with Martino Lupini and Aristotelis Panagiotopoulos.