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.
British-born Abedi had a "relatively minor" criminal record as a teenager but was not known to police for holding extremist views.
He appeared on police logs in 2012 over offences of theft, receiving stolen goods and assault, Greater Manchester Police Chief Constable Ian Hopkins said.
However, Abedi had not featured in Prevent, the Government's voluntary counter-radicalisation scheme.
Mr Hopkins also said charges of conspiracy to murder could be brought as a result of the huge investigation into Abedi's suspected network.
Twenty-two people were killed and dozens of others seriously injured when the bomber struck after a concert by American singer Ariana Grande.
Justin Bieber, Katy Perry, Coldplay and a host of other international music stars will join Grande at a benefit concert for victims on Sunday.
On Tuesday night Liam Gallagher played his first solo gig in Manchester, with profits donated to the families of the terror attack victims.
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.