Brad Klee wrote: ----- I think it would help immensely to get into details and see worked examples. ----- Agreed. I didn't want to make that post too long. ----- For a complexified algebraic curve, what is the algorithm to construct a detailed tiling of the uniform time domain? (You can use triangularization if you want) ----- I'm not aware of any such algorithm. But I'm not sure if I understand exactly what is meant by "complexified algebraic curve". ( There is a very nice class of surfaces known as "complex projective curves", which are smooth surfaces defined by the locus of common zeroes of a finite family of n-1 equations P_j(Z_0, Z_1, ..., Z_n) = 0 on complex projective n-space CP^n, where each P_j is a homogenous polynomial with complex coefficients. For example, the spectacularly beautiful "Klein quartic" is defined as K = {(X : Y : Z) ∊ CP^2 | P(X,Y,Z = 0} where P(X,Y,Z) = X^3 Y + Y^3 Z + Z^3 X. K is topologically a compact orientable surface of genus 3. The group of orientation preserving isometries of K is the simple group of order 168, and it admits a tiling by 24 regular hyperbolic heptagons, 3 per vertex, compatible with these isometries. ) Or what is meant by "the uniform time domain". ----- By detailed I mean that singular points and algebraic cycles should be marked. To step up a dimension, just ask what happens when the curve/surface depends on a shape parameter. If you start with a genus g=0,1, or 2 surface and deform it continuously via the shape parameter, how can the resulting 3-manifold be described in terms of the Thurston Geometries? So yes, it’s more than I want to know, but also less than what I want to know. For me (and many others), more constructive methods = less confusing formalism. ----- How is a continuous deformation of a surface defined? If M_g is your initial surface, will the resulting 3-manifold always be topologically M_g x [0,1] ? My previous post was exclusively about topology, and not at all about geometry. —Dan