The body of Jerry Appicella, aged 51, was found in his home on Craganour Place, Denaby Main, on Sunday after police received calls of concern for his welfare.
Detectives believe he was involved in an incident with a group of people in an alleyway leading to Hickleton Street in Denaby Main on Tuesday, December 3.
He then walked home and was found dead 12 days later.
Three teenage girls and one teenage boy have been arrested on suspicion of murder as part of the police probe.
South Yorkshire Police said: “Enquiries are ongoing and officers are keen to speak to anyone with information which might help them piece together the circumstances around Mr Appicella’s death.
“We again would like to offer our reassurance that we believe this to have been a targeted attack and there is no wider risk to the community.”