A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces

Abstract

Using a result of Harer, we prove certain upper bounds for the homotopical/cohomological dimension of the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most $k$ rational components. These bounds would follow from conjectures of Looijenga and Roth-Vakil.