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

Details The Fine Structure of the Kasparov Groups II: topologizing the UCT The Fine Structure of the Kasparov Groups II: topologizing the UCT

2001-07-10

arxiv.org/abs/math/0107071v1

Mathematics

Operator Algebras

25 pages

Scientific paper

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 J. Rosenberg and the author, as well as its canonical three term decomposition which exists under bootstrap hypotheses. We show that the various topologies on Ext_{\Bbb Z}^1(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 Pext_{\Bbb Z}^1(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.

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