Definition of "Representing" a Handlebody (Lefschetz Fibration)? Sorry, I could not find a clear explanation of the meaning of the word represented in the following:"any 4-dimensional 2-handlebody W can be represented by a topological (achiral) Lefschetz fibration over the disk", in the first paragraph of the 1st page paper: http://arxiv.org/pdf/1104.4536v3.pdf
I can't see what form the representation  takes in here. Also, if someone could please help me figure out, at the bottom of the same 1st  page, in "Theorem A" , the meaning of "2-equivalent" 4-D handlebodies. I know what a 4-D 2-handlebody is, but I don't know the meaning of "2-equivalent". Thanks for any help.

In that paper, page 358, 2-equivalence is the equivalence relation on four-dimensional 2-handlebodies generated by 2-deformations, meaning handle isotopy, handle sliding and addition/deletion of cancelling pairs of handles of indices less than or equal to 2.
so, four-dimensional 2-handlebodies are 2-equivalent if and only if they are in the same equivalence class.
For more helpful introduction, I would like to recommend you to read the third author's lecture notes.
