Police launched an investigation after the body of Jonathan Scotland, 33, of Glasgow, was found outside Blythswood Court, in the Anderston area of the city, by a member of the public on Wednesday.
The incident sparked an alert as police cordoned off the area around Cadzow Street.
Police said that the man, a former pupil at Lenzie Academy, had fallen eight floors to his death.
An incident tent was set up in the street below the public walkway leading from Blythswood Court's Davaar block to the other two high rise buildings in the housing complex.
Uniformed officers as well as plain clothed police searched the scene and carried out door to door inquiries.
As well as Blythswood Court, the busy area is home to commercial businesses.
At the time of the incident, local residents spoke of their shock, and told how they heard screams after a fracas on the 8th floor.
One said he heard police "kicking in" the door of a flat
It is thought Mr Glasgow fell from a window off a sitting room.
It is understood the flat at the centre of the police probe has been occupied by a single woman, aged 28, for the past nine months.
Another resident said he later saw the shoeless victim lying on the street and surrounded by debris.
Raymond Collings, 59, said: "At about half-past-one, I heard banging and shouting.
"The next thing I heard were terrible screams. They were animal-like, I have no idea if it was a man or a woman, it was a terrible sound."
James Kyle, 71, said: "I could clearly hear a fracas but I didn't think much about it because there has been a lot of that at this flat across from me.
"But about half-an-hour later, I heard the police outside on the landing, and the next thing is the door was being kicked in, forced."
Other locals said men were frequent visitors to the flat and that the woman who occupied it may have been homeless before she was allocated the property.
Local resident Marion Evans, 48, said: "The lassie has been in the flat for about nine months. It is no secret that there have been a lot of boys visiting her."
Others said the incident was just the latest in a series of problems which have plagued the block and made life a misery for residents.
There have been reports of drug users shooting up in stairwells, fire-raising, and letter box thefts, which have forced some to consider leaving the homes they have lived in for more than 20 years.
It had been suggested last year that a reduced concierge service and a high turnover of tenants were causing problems with anti-social behaviour.
Homeowners and tenants launched the Blythswood Residents' Association at the end of 2011 to try to fight back.
The complex, built in the 1960s, originally belonged to the Glasgow Housing Association (GHA), but residents voted to transfer the housing stock to Glasgow West Housing Association in 2011
A Police Scotland spokeswoman said: "Mr Scotland's death is being treated as suspicious and enquiries are continuing."