Greater Manchester Police have confirmed the name of the suspect suicide bomber as Salman Abedi, 22.
Greater Manchester Police said the priority was to establish whether Abedi had worked alone or not.
He is thought to have blown himself up in the arena's foyer shortly after 10:30pm on Monday, as fans were beginning to leave a concert by US singer Ariana Grande.
Abedi, who had at least three siblings, had lived at several addresses in Manchester, including a property at Elsmore Road, Fallowfield, which was earlier raided by police.
Greater Manchester Police Chief Constable Ian Hopkins said searches at two addresses had been carried out, including the one in Fallowfield, where a controlled explosion had been used to gain "safe" access.
He said Abedi had not been formally identified and so would not comment further.
Mr Hopkins passed on "heartfelt sympathies to all the innocent people caught up in last night's despicable act", adding that specially-trained family liaison officers were supporting families.
"There has been much speculation and names of those who may have been killed in the media and social media," he said.
"We accept that this is inevitable, however, we ask that people allow the police and coroner to release the names once the families are ready and appropriately supported."