## Logic Seminar

This talk will present work of Bergfalk, Lupini, and Panagiotopoulos. Given a nice enough space $X$ and a nice homology theory, the homology groups of $X$ may be viewed as topological groups, namely as quotients of a Polish abelian group by a Polishable subgroup. But topological structure of homology groups are rarely studied because the quotient topology might be bad, in other words, the category of Polish abelian groups is not abelian. In this talk we define a definable version of Steenrod homology for compact metrizable spaces which we can use to talk about topological structures of homology groups. The definable homology of a compact metrizable space will be an object in a minimal abelian extension of the category of Polish abelian groups. In the second talk we will see how this topological information makes definable homology a strictly finer invariant than classical Steenrod homology for solenoids.