The Fine Structure of the Kasparov Groups II: Topologizing the UCT

The Kasparov groups KK*(A,B) have a natural structure as pseudopolonais groups. In this paper we analyze how this topology interacts with the terms of the Universal Coefficient Theorem (UCT) and the splittings of the UCT constructed by Rosenberg and the author, as well as its canonical three term decomposition which exists under bootstrap hypotheses. We show that the various topologies on ExtZ1(K*(A),K*(B)) and other related groups mostly coincide. Then we focus attention on the Milnor sequence and the fine structure subgroup of KK*(A,B). An important consequence of our work is that under bootstrap hypotheses the closure of zero of KK*(A,B) is isomorphic to the group PextZ1(K*(A),K*(B)). Finally, we introduce new splitting obstructions for the Milnor and Jensen sequences and prove that these sequences split if K*(A) or K*(B) is torsion-free.