In "The Crucible" what are the turning points in which the relationship of John Proctor and Abigail Williams changes?
In the beginning of the play, a turning point occurs when Abby tries to put the moves on John in Act One, and he firmly rejects her. He tells her, "We never touched," and lets her know that he is trying to move on and leave her behind in the process. Abby is infuriated by that turn of events, and lashes out against Elizabeth as the reason for it. They are interrupted in their conversation, but the gist is that John tries to stomp out any hope that Abby had.
Later, in Act Two, another turning point occurs when John realizes that Abby is willing to kill off his wife, Elizabeth, in order to be with him. Elizabeth is arrested in this act, through trickery on Abby's part, and John must have had the grave realization that she was trying to get him back.
In Act Three, , in the extra scene that is sometimes taken out, John gives her hope again when he summons her to meet him the forest. But, it soon becomes clear that he hasn't met her there to renew their relationship. She is still hopeful; in fact, she admits that she looks forward to being his wife when "the world is white again." He rejects this though, and shows true disgust for her, which makes her angry. Later, a true turning point occurs when John accuses Abby of being a whore, right in front of everyone in the courts. She is astounded that he would be willing to confess to adultery in front of everyone--she had counted on him not doing that. He surprises her and catches her off guard, and she finally realizes just how serious he is about saving Elizabeth and proving the girls are frauds. She holds her ground though, and refuses to admit that she participated in the affair. John's feelings for her have turned to pure hatred in this act, and she retaliates by turning on Mary, who in turn turns on John, leading to his arrest. Whether Abby intended that or not is unclear, but it was the unfortunate result of her refusing to admit to any wrongdoing.
Throughout the course of the play, the relationship between John and Abby changes from a love affair, to tortured longing, to hatred and betrayal. I hope that helped; good luck!