  We study the noncommutative topology of the $C^*$-algebras
$C(\mathbb{C}P_q^{n})$ 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 $\mathbb{C}^{n+1}$. 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.

Operator Algebras;K-Theory and Homology;Quantum Algebra