Frank McGreevy, 51, a former republican prisoner, had been on a life support machine following Saturday's attack in Ross Street. He died in hospital.
A 20-year-old man, who handed himself into police, remains in custody.
Mr McGreevy was found injured by his 15-year-old son, Francis. He also had another son, Tiernan, who is nine.
It is not clear if Mr McGreevy was attacked after confronting young people about anti-social behaviour.
The PSNI have said that the man who is being questioned handed himself in to Grosvenor Road police station.
Police have appealed for anyone who was part of a group outside Mr McGreevy's flat at about 1830 GMT to contact them.
Sinn Fein and SDLP politicians met senior officers at Grosvenor Road police station on Monday.
Afterwards, the politicians called for a broad community-based approach to tackling the problems in the area.
Meanwhile, a vigil has been held in west Belfast to remember Mr McGreevy.
Earlier on Tuesday, police were given a further 36 hour extension to question the 20-year-old man who has been arrested.
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