I came across Generalizing polygon file while maintaining topology in QGIS? which asks for a QGIS solution to my exact question. I'm trying to do this in ArcMap. I start with a polygon shapefile (WGS84).
I verified by importing it into a file geodatabase feature dataset topology that it does not have any gaps or overlaps. Then I run the Simplify Polygon ArcToolbox tool on it.
The output has visible gaps and overlaps between some polygons. I added that layer to the topology and it shows me the rule-violating areas.
I'm kinda an ArcGIS Topology noob though - one thing that has me confused is the Topology Toolbar remains entirely disabled. And even if that wasn't a problem I'm not keen to have to repair gaps and overlaps one by one interactively. Even if there aren't that many issues this time, there will be next time. Ideally I'd come up with a simplification with topology preservation workflow that is entirely automated. Is there a better way?