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The moduli spaces of compact and connected Riemann surfaces has been acentral topic in modern mathematics in recent years. Thus their homologicaldimensions become important invariants. Motivated by the emergence mathematicalcounterparts of open-closed string theory, we give an estimate of rationalhomological dimension of Riemann suraces with possible boundary and markedpoints(can lie on both interior and boundary). We hope it will haveapplications in open-closed theory, for example, open-closed Gromov-Wittentheory in the future.

A Very Simple Estimate Of Rational Homological Dimension Of Moduli Spaces Of Riemann Surfaces With Boundary And Marked Points