Triangulations of surfaces with boundary and the homotopy principle for functions without critical points
The author considers triangulations of compact surfaces with boundary and marked points. This is connected with the moduli space of Euclidean structures on the sphere with conical singularities, bifurcation diagrams of quadratic differentials, Teichmüller space and complexity of fiber bundles. Allowing that a vertex may belong to an edge or a face, a more general definition of triangulation is given. Basic transformations called flips are defined and it is shown that two triangulations are connected by means of a finite number of flips (flip equivalence). Later on, an additional structure, marked vertices, are added to triangulations and it is shown that the flip equivalence need not always be satisfied. After considering relations between some spaces and their connected components, and then defining homotopy equivalence between these spaces, the author associates a generic function to each triangulation. In that way homotopy classification of functions not having critical points is reduced to the description of triangulations with marked points up to homotopy and flips.
scientific article; zbMATH DE number 1333261