Critical levels and Jacobi fields in a complex of cycles

In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which are explicitly constructed and shown to give a complete description of the entire space of tight geodesics. Jacobi fields measure the extent to which geodesic stability breaks down. Unlike most finiteness properties of curve complexes, the arguments presented here do not rely on hyperbolicity, but rather on structures similar to Morse theory.