An equivariant deformation retraction of the Thurston spine

TL;DR

This paper constructs a mapping class group-equivariant deformation retraction of Teichmüller space onto the Thurston spine, a cell complex capturing hyperbolic surfaces with systoles that cut into polygons, linking geometric and topological structures.

Contribution

It introduces a new equivariant deformation retraction of Teichmüller space onto the Thurston spine, aligning geometric properties with the complex's cell structure.

Findings

Deformation retraction is equivariant under the mapping class group.

The image is contained within Thurston's original spine complex.

The dimension matches the virtual cohomological dimension of the mapping class group.

Abstract

This paper shows that there is a mapping class group-equivariant deformation retraction of the Teichm\"uller space of a closed, orientable surface onto a cell complex of dimension equal to the virtual cohomological dimension of the mapping class group. The image of the deformation retraction is contained in the CW complex first described by Thurston -- the Thurston spine. The Thurston spine is the set of points in Teichm\"uller space corresponding to hyperbolic surfaces for which the set of shortest geodesics (the systoles) cuts the surface into polygons.