The plot of the book is set in motion when Sam Spade's partner is killed, along with the man he was hired to shadow. Miles Archer is Spade's partner; he is described as a "louse" and a "lecher" and is shot by Brigid. Archer was following Floyd Thursby, who was helping Brigid; Thursby is shot by Wilmer as a scare tactic to Brigid. These two murders are initially linked to Spade, as he was having an affair with Archer's wife and the two shootings were close together. Later in the story, after a lot of things about Brigid and the Maltese Falcon have been revealed, Captain Jacobi, who smuggled the Falcon for Brigid, is shot by Wilmer, but manages to get the (fake) Falcon to Spade before he dies.
"Anyways we got it from Cairo. Gutman's dead. The kid had just finished shooting him up when we got there."
Spade nodded. "He ought to have expected that," he said.
(Hammett, The Maltese Falcon, Google Books)
Finally, Wilmer shoots Casper Gutman, the rich man who had been searching for the real Falcon all this time. Wilmer's betrayal comes after Gutman shows willingness to frame Wilmer so the police will have someone to arrest for the murders. This makes four murders in all, three of which are directly linked to Brigid's involvement. Two of the murders (Archer and Thursby) are attempts to manipulate others, while Jacobi's murder was one of convenience and Gutman's murder was one of passion.