Basic partitions and combinations of group actions on the circle: A new approach to a theorem of Kathryn Mann

Let $Pi_g$ be the surface group of genus $g$ ($ggeq2$), and denote by $RR_{Pi_g}$ the space of the homomorphisms from $Pi_g$ into the group of the orientation preserving homeomorphisms of $S^1$. Let $2g-2=kl$ for some positive integers $k$ and $l$. Then the subset of $RR_{Pi_g}$ formed by those $varphi$ which are semiconjugate to $k$-fold lifts of some homomorphisms and which have Euler number $eu(varphi)=l$ is shown to be clopen. This leads to a new proof of the main result of Kathryn Mann cite{Mann} from a completely different approach.