  We establish combinatorial versions of various classical systolic
inequalities. For a smooth triangulation of a closed smooth manifold, the
minimal number of edges in a homotopically non-trivial loop contained in the
$1$-skeleton gives an integer called the combinatorial systole. The number of
top-dimensional simplices in the triangulation gives another integer called the
combinatorial volume. We show that a class of smooth manifolds satisfies a
systolic inequality for all Riemannian metrics if and only if it satisfies a
corresponding combinatorial systolic inequality for all smooth triangulations.
Along the way, we show that any closed Riemannian manifold has a smooth
triangulation which "remembers" the geometry of the Riemannian metric, and
conversely, that every smooth triangulation gives rise to Riemannian metrics
which encode the combinatorics of the triangulation. We give a few applications
of these results.

Geometric Topology;Combinatorics;Differential Geometry;Metric Geometry