Given a finite set $A$ on the Riemann sphere and a homeomorphism $f$, may I say there exists a quasiconformal homeomorfism isotopic to $f$ relative to the set $A$?



Quasiconformal deformation
Yes, as long as $f$ is orientation-preserving. 


By a theorem of Munkres-Smale-Whitehead (see Corollary 1.18 in Manifolds with Transverse Fields in Euclidean Space, by Whitehead), $f$ can be approximated arbitrarily well by smooth diffeomorphism. 


Let $g$ be a diffeomorphism close enough to $f$. We can modify $g$ in a neighborhood of $A$ so that $g|\_A=f|\_A$. First pick disjoint disks about the points of $f(A)$ each containing a point of $g(A)$. Then, given two points in the interior of a closed disk, it is easy to construct a smooth map which agrees with the identity on a neighborhood of the boundary and sends the first point to the second point.


In the end we have a smooth map $g$ (hence a quasiconformal one since the sphere is compact) with $g|\_A=f\_A$ as close as we want to $f$. If we choose $g$ close enough, then it will be isotopic to $f$ rel the set $A$.


