  The moduli spaces of compact and connected Riemann surfaces has been a
central topic in modern mathematics in recent years. Thus their homological
dimensions become important invariants. Motivated by the emergence mathematical
counterparts of open-closed string theory, we give an estimate of rational
homological dimension of Riemann suraces with possible boundary and marked
points(can lie on both interior and boundary). We hope it will have
applications in open-closed theory, for example, open-closed Gromov-Witten
theory in the future.
