How does the handkerchief function in act 4? Keeping track of its movements from here to the end of the play.

Othello's handkerchief that he had given to
Desdemona before the play begins becomes the play's most important prop and an ironic
symbol of their relationship.  The handkerchief actually comes into play in Act 3 when
Desdemona attempts to wipe the brow of her agonized husband with it.  In his impatience,
he brushes it away, and since Othello is distraught at the possibility of Desdemona's
unfaithfulness and Desdemona is worried about the strange behavior of her husband,
neither notices that the handkerchief is dropped.  It is this "trifle" that Iago uses to
frame Desdemona. Emilia picks up the handkerchief for her husband, and Iago plants it in
Cassio's lodging. 

 In Act 4, the handkerchief reappears in
the hands of Bianca, Cassio's mistress.  Cassio had found the handkerchief in his room
and given it to Bianca to copy the work. Bianca, thinking it the token of another
mistress,  gives it back to Cassio.  This scene in enacted while Othello is
eavesdropping.  Othello incorrectely believes that the handkerchief is the "ocular"
proof that Cassio is sleeping with Desdemona. 

After this
scene, the handkerchief is not mentioned again until Act 5 when after Desdemona's death,
Othello defends his act by declaring that Desdemona was unfaithful, and that the
handkerchief in Cassio's hands was proof.  Emilia realizes now the truth.  She explains
that it was she who took the handkerchief and gave it to her husband.  Iago, not
Desdemona, gave the handkerchief to Cassio.

This revelation
clears Desdemona's name, and Othello now is full of regret for the wrongful act he has
committed.