Understanding the well-rounded deformation retraction of Teichm\"uller space

Abstract

In [10] it was shown that there is a mapping class group-equivariant deformation retraction of the Teichm\"uller space of a closed surface onto a CW complex with dimension equal to the virtual cohomological dimension of the mapping class group. This paper studies the image of this deformation retraction and shows that when the analogy with the well-rounded deformation retraction of $SL(n,\mathbb{Z})$ is defined correctly via a notion of duality, this deformation retraction is analogous to the well-rounded deformation retractions of [2], [24] and [26]. In the process, an elementary necessary condition is derived for a cycle in the geometric realisation of Harvey's curve complex to represent a nontrivial homology class.