This is the first talk in a three-part series in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information.
In the first talk we will focus on the "definable enrichment" of the first derived functors of Hom(−,−)Hom(−,−) and lim(−)lim(−). We will show that the resulting "definable Ext(B,F)Ext(B,F)" for pairs of countable abelian groups B,FB,F; and the "definable lim1(A)lim1(A)" for towers AA of Polish abelian groups substantially refine their purely algebraic counterparts. In the process, we will develop an Ulam stability framework for quotients of Polish groups GG by Polishable subgroups HH and we will provide several rigidity results in the case where the ambient Polish group GG is abelian and non-archimedean. A special case of our rigidity results answers a question of Kanovei and Reeken regarding quotients of the pp-adic groups.
This is joint work with Jeffrey Bergfalk and Martino Lupini.