  Quantum Steiffel manifolds were introduced by Vainerman and Podkolzin in
\cite{VP}. They classified the irreducible representations of their underlying
$C^*$-algebras. Here we compute the K groups of the quantum homogeneous spaces
$SU_{q}(n)/SU_{q}(n-2), n\ge 3$. Specializing to the case $n=3$ we show that
the fundamental unitary for quantum $SU(3)$ is nontrivial and is a unimodular
element in $K_1$.
