The Differential Topology of the Thurston Spine of Teichm\"uller Space

In \cite{Thurston}, a short, simple and elegant construction of a mapping class group-equivariant deformation retraction of Teichm\"uller space of a closed compact surface was given. The preprint \cite{Thurston}, which unfortunately is not online, has not been broadly accepted. The purpose of this paper is to go through the construction in detail, resolve any questions that have arisen in the literature and in personal communications. An explicit example is given to show that one of the claims needs to be modified, and details of how to do this are given. A mapping class group-equivariant deformation retraction of the Thurston spine onto a complex of dimension equal to the virtual cohomological dimension of the mapping class group is then constructed.