Mark Conway, 26, was discovered at a property in the Milton area of Glasgow on Wednesday morning, less than an hour after he was seen walking several streets away.
The victim, from Thornliebank, East Renfrewshire, and who was known as Keddy to friends, died at the scene.
Detective Chief Inspector Jim Downie, of Strathclyde Police, said: "We know that Mr Conway was walking on Skirsa Street in the Cadder area around 7:25am and a short time later he was found seriously injured within the flat in Scaraway Street.
"However, at this stage we do not know exactly where, or how, the young man came about his injuries. Officers are carrying out door-to-door inquiries and scouring CCTV footage to try to piece together his movements prior to being found."