Definition of "Representing" a Handlebody (Lefschetz Fibration)?

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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.