7.1 Characterizing genus changes at criticalities.

At critical points, the topology changes are easy to characterize. Critical sets (which may include ridges and more elaborate structures) pose different issues. However, for maximum critical sets, the topology of the associated object created can be computed from the structure of the connected set of readings S. In particular, the genus can be determined by constructing the boundary of this component of $\bf {X}_{f}^{\geq \delta ( S )}$. For example, one can use SpiderWeb on the set and use Euler's law. Similarly, for minimum sets and saddle sets, the topology change is based on the number of connected components of Highs that exist among the neighbors. One doesn't need special purpose algorithms for each type of criticality or for each dimension, derived from differential geometry or differential topology, once one has reduced the questions to basic combinatorial issues.

Basically, the topology change at a critical set value is determined by the number of distinct High components (among the neighbors) that are connected by the critical set for saddles and minima, and by the genus of the boundary manifolds for maximal sets. This takes the place of the Morse data, in our Digital Morse Theory. We have uniformly characterized all possible critical structures (the so-called ``zoo'' of criticalities of non-Morse functions) using basic combinatorial methods.