Nine people are currently being held
British police say they have "got our hands around" a large part of the network connected to Manchester Arena bombing Salman Abedi.
Senior counter-terror officer Mark Rowley said "immense progress" had been made in the investigation, but further arrests were likely.
Nine people are currently being held by police in connection with the bombing, which killed 22 people.
"They are very significant, these arrests," Assistant Commissioner Rowley said.
"We are very happy we've got our hands around some of the key players that we are concerned about but there's still a little bit more to do."
Police have raided a number of businesses and homes in the hunt for those who may have helped Abedi.
Assistant Commissioner Rowley sought to reassure people planning to head out to events in the UK this weekend.
"Go out as you planned. Enjoy yourselves and be reassured by the greater policing presence you will see," he said.
"We can't let the terrorists win by dissuading us from going about our normal business."
Police have reviewed security at more than 1,300 events over a weekend that includes a number of football matches at Wembley and an athletics meeting in Manchester.
Despite the apparent success of the investigation, the terrorism threat level remains at critical.
"Even with all the progress made so far, there still remain important lines of enquiry that will further build our confidence and understanding of the events that led to this ghastly attack," Assistant Commissioner Rowley said.
"That will enable those who set the threat level to revise their judgement."