This is one of those issues in that was in Exchange 2013 RTM is is still there in CU8. I'm pretty sure it's very easy to fix and Microsoft PSS is aware of the issue, but anyway this is the error I'm talking about:
The Microsoft Exchange Health Manager service depends on the following service: MSExchangeADTopology. This service might not be installed.
This of course makes perfect sense because the Microsoft Exchange Active Directory Topology Service is not installed on an Edge Transport server. So in order to remove the dependency on the MSExchangeADTopology service we need to fire up Regedit and navigate to the following key: HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\MSExchangeHM. Open the Value DependOnService and remove MSExchangeADTopology from the Value Data.
Make sure the only remaining entry is eventlog and click OK to save. Reboot your server and verify that the Microsoft Exchange Health Manager is running.