M _к -Polyhedral Complexes of Bounded Curvature

Abstract

In this chapter we return to the study of metric polyhedral complexes, which we introduced in Chapter I.7. Our focus now is on complexes whose curvature is bounded from above. The first important point to be made is that in this context one can reformulate the CAT(κ) condition in a number of useful ways. In particular, Gromov’s link condition (5.1) enables one to reduce the question of whether or not a complex supports a metric of curvature ≤ κ to a question about the geometry of the links in the complex. This opens the way to arguments that proceed by induction on dimension, as we saw in (I.7). If the cells of the complex are sufficiently regular then the link condition can be interpreted as a purely combinatorial condition; in particular this is the case for cubical complexes and many 2-dimensional complexes.