  Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we
show that (un-)boundedness of the composition operator ${\bf C}_f$ on the
spaces $Q_{\alpha}(\mathbb R^n)$ depends on the index $\alpha$ and the
degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of
the index $\alpha$ and the local/global self-similar Minkowski dimension of the
degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also
reveals a completely new phenomenon, which is totally different from the known
results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently,
Tukia-V\"ais\"al\"a's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of
an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is
shown to preserve $Q_{\alpha} (\mathbb R^n)$ for any $(\alpha,p)\in
(0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_{\alpha}(\mathbb R^n)$ is
shown to be invariant under inversions for all $0<\alpha<1$.
