We have a feature data set with three polyline feature classes that share geometry and participate in a topology with just one rule. These can be edited in ArcMap using the Topology Edit tool, where a user modifies one feature’s geometry and the features in the other two feature classes with coincident geometries snap in place along with the feature being edited. We found ways in the API to create and edit topologies and to check for topology errors, but not a way to edit while maintaining shared geometry like ArcMap’s topology edit tool. Our Esri representative spoke to some developers at the user conference who said the nuts and bolts exist. We are reaching out for a description of the methodology to simulate what the Topology Edit tool does because without it, our locally-hosted application users would have to snap every vertex in place manually in order to honor the topology rule "must be covered by feature class of".
Amending this post - my Esri rep directed me to this forum but I now see this is for JavaScript API, not Java. We are using the Java API because we are working with versioned data which is required in order to work with a topology, and versioning with the "move edits to base" option will not work per our tests.
