Salman Abedi bought most of the key component parts of the suicide bomb he detonated in the Manchester Arena terror attack, police said.
Many of his movements and actions in the four days after his return to the UK from Libya leading up to the May 22 atrocity were also carried out alone but detectives have not ruled out that he was part of "a wider network".
Eleven people remain in custody for questioning following a series of raids across the country as three men - two aged 20 and 24 from the Fallowfield area and a 37-year-old from Blackley - were released without charge on Tuesday.
Updating the "huge progress" made in the inquiry, Detective Chief Superintendent Russ Jackson, head of the North West counter terrorism unit, said: "Much of the investigation has been painstakingly working through Salman Abedi's last movements. We have done this by examining his movements on CCTV and other interactions he has had, whether it be with people or the phone calls he has made.
"With specialist support we have also have a good understanding of the likely component parts of the bomb and where these came from.
"Our inquiries show Abedi himself made most of the purchases of the core components and what is becoming apparent is that many of his movements and actions have been carried out alone during the four days from him landing in the country and committing this awful attack.
"It is vital that we make sure that he is not part of a wider network and we cannot rule this out yet. There remain a number of things that concern us about his behaviour prior to the attack and those of his associates which we need to get to the bottom of."
Mr Jackson said police were "especially keen" to find out why Abedi, 22, kept going back to the Wilmslow Road area of the city as they continue to attempt to trace a blue suitcase he used during those trips.
Although detectives have no reason to believe the luggage contains anything dangerous, they are warning the public not to approach it and instead call 999 immediately.
A police cordon has been placed around Banff Road in Rusholme, off Wilmslow Road.
In a statement, Greater Manchester Police said: "We are currently searching a property on Banff Road in Rusholme in relation to last Monday's attack at Manchester Arena.
"There is a cordon in place. Please avoid the area."
Police had previously searched the property in Rusholme's Curry Mile on Monday.
A bomb disposal unit arrived at the scene surrounding Banff Road but is said to be a precautionary measure.
Police previously said they knew that Abedi had visited the Wilmslow Road area with the missing blue suitcase in the lead-up to the attack.
They appealed for sightings of Abedi with the case between May 18 and 22, and released an image of him with the luggage taken from CCTV in Manchester city centre on the day of the suicide bombing.
Anybody with information about the whereabouts of the case was urged to call the anti-terrorist hotline in confidence on 0800 789 321.
The bomb disposal unit later left but the cordon remained in place.