Triangulations of Surfaces

This paper uses an elementary surgery technique to give a simple topological proof of a theorem of Harer which says that the simplicial complex having as its top-dimensional simplices the isotopy classes of triangulations of a compact surface with a fixed set of vertices is contractible, except in a few special cases when it is homeomorphic to a sphere. (Triangulations here are allowed to have triangles with coinciding vertices or edges.) The proof yields mild generalizations of Harer’s theorem, allowing more general vertex sets, as well as extending to a larger complex whose simplices correspond to curve systems consisting of circles as well as arcs. As a corollary we deduce the well-known classical fact that any two isotopy classes of triangulations of a compact surface with a fixed set of vertices are related by a finite sequence of elementary moves in which only one edge changes at a time.