| Gromov's systolic estimate is considered one of the deepest results in systolic geometry. It states that, for an essential Riemannian n-manifold M, the length of the shortest noncontractible loop, or systole, of M, denoted syspi1 ( M) satisfies syspi1 (M) ≤ Cn n√Vol(M) where the constant Cn only depends on n and not on M. We will prove a discrete version of related theorems for triangulated surfaces. The argument involves creating a special Riemannian metric on a triangulated surface whose total volume is close to the number of facets in the triangulation. This metric then allows one to convert Riemannian geodesics to homotopic edge paths of controlled length. The proof of the analogous inequality in the case of a triangulated triangles then follows easily from these facts.;We then apply our discrete version to facts about triangulations of orientable surfaces. Given a triangulated, orientable, closed surface with x 2-simplices, we can ask how many 3-simplices are required to "fill" the triangulation: that is, produce a triangulated 3-manifold whose boundary triangulation is the triangulated surface with which we started. Our method produces such a 3-manifold with no more than O(x log2 x) simplices.;We will also prove that a discrete version of this inequality implies the Riemannian version. The proof of this fact involves creating a triangulation of a Riemannian manifold that is in some sense aware of the geometry of the manifold. We embed M in Rm using the Nash Embedding Theorem and use an argument of Whitney's to produce a triangulation whose simplices are large in volume relative to their edges in the metric of Rm. By working on a small enough scale, one obtains information about the geometry of the simplices embedded in M in the induced path metric. Since M is isometrically embedded, this gives the result. Again, once this obtained, proving that a discrete systolic inequality implies a Riemannian one is quite simple.;Finally, we present Whitney's argument and the necessary modifications needed to obtain our triangulation theorem for embedded submanifolds of Rm. |
