Split extensions and KK-equivalences for quantum projective spaces

We study the noncommutative topology of the C*-algebras C(CP n q ) of the quantum projective spaces within the framework of Kasparov’s bivariant K-theory. In particular, we construct an explicit KK-equivalence with the commutative algebra C n+1 . Our construction relies on showing that the extension of C∗ -algebras relating two quantum projective spaces of successive dimensions admits a splitting, which we can describe explicitly using graph algebra techniques