The mapping class group of 2-dimensional surfaces (of genus g) admits as a classifying space the moduli space M_g. This moduli space also happens to classify various structures of interest that one can consider on a surface (for ex. complex structures, conformal structures, Riemannian metrics,..), and a geometric model for this classifying space is obtained as the quotient of the Teichmuller space by the mapping class group.
In joint work with Dan Petersen we described a model for the classifying space of the mapping class group of 3-dimensional handlebodies as an explicit open subspace of M_g.
After introducing the objects of interest and motivating their study, I will discuss some consequences of our construction and, time permitting, I will present some ideas of the proof that this space is indeed a classifying space.