I've been looking for the same solution: editing PostGIS data with ArcMap Desktop. ESRI sales rep says ArcGIS for Server is needed.
There is this solution ST-Links SpatialKit http://www.st-links.com/Pages/default.aspx It can be installed as an extension.
I added a PostGIS layer, started editing, reshaped the feature, and saved. Tested with PostgreSQL 9.3, PostGIS 2.1 and ArcMap 10.3.1 with their free trial. However, I was unable to consistently get the data to save back into PostGIS. Still working through the manual.
While this doesnt use SDE, it is not free either, but the pricing seems fair to the ESRI ArcGIS for Server software.
Not sure how to deal with topology checks, but I suspect it will have to be on the backend with PostGIS topology functions.