Isotopic triangulation of a real algebraic surface

International audienceWe present a new algorithm for computing the topology of a real algebraic surface $S$ in a ball $B$, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to $S$, and even a simplicial complex isotopic to $S$ by exploiting properties of the contour curve of $S$. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of $S$, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of $S$ from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accomodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^{n,0}(V) spheres S^n.Comment: 35 pages, 9 figures, final version to appear in Topolog

Non-geometric veering triangulations

Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic metric with all tetrahedra positively oriented. This paper describes a computer program Veering, building on the program Trains by Toby Hall, for generating these triangulations starting from a description of the homeomorphism as a product of Dehn twists. Using this we obtain the first examples of non-geometric veering triangulations; the smallest example we have found is a triangulation with 13 tetrahedra

Simple Loops on Surfaces and Their Intersection Numbers

Given a compact orientable surface $\Sigma$, let \Cal S(\Sigma) be the set of isotopy classes of essential simple loops on $\Sigma$. We determine a complete set of relations for a function from \Cal S(\Sigma) to $\bold Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in \bold R^{\Cal S(\Sigma)} and P(\bold R^{\Cal S(\Sigma)}) defining Thurston's space of measured laminations and Thurston's compactification of the Teichm\"uller space. These equations are not only piecewise integral linear but also semi-real algebraic.Comment: 42 pages, 29 figure