I own a flat with shared access front door and pay an annual service charge, I have been told I need a smoke alarm linked from inside to the outside (exit door) of the flat,that is at the top of a stair well, what I want to know is am I responsible for the alarm at the top of the stairwell? Or is my management company responsible? 

It sounds like as a result of a fire risk assessment your block has determined that a full evacuate policy is now required and as part of that the communal fire alarm system needs to be linked into each flat so it will activate before the escape route is affected and be loud enough in the flat to wake people.

Whilst this new element of the system is inside the flat it is still part of a communal system required for Fire Safety Order compliance and thus the responsibility of the freeholder's agent through the service charge. You would be expected to give reasonable access to let their contractor test your detector, which would only be once a year.