The co-worker who lent me this didn't have volume two. It seems that all I missed was a human kid obsessed with Tiphares, the awesome city that floats above the Earth, and then his body got destroyed and he became a cyborg. Then Alita turned him into her pet project and he killed himself. So that's that.
Alita was so traumatized, as she should be, so she runs away to drown her pain in Rollerball. She's the best player in the third division. Ido finds her after a long search and because she doesn't acknoledge him, he decides to destroy her pride so she'll come back to him. What?
There's a lot of fast pace, arm ripping, panther kunst, blading action. All so Alita can become second class and fight the best player of that division, Jashugan. Who happens to be hanging out with Ido.
The climax of the book is Alita and Jashugan arm wrestling while talking about the power of Chi. He blows her arm with his Chi. This to Alita is like foreplay and even though Jashugan let her win you can feel the cyborgy sexual tension as his arm falls from his shoulder.
They are meant to be.