Restricting Base of Bundle. Which Properties are Preserved?

WWGD

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Hi All,
Please let me set things up before the actual question:
I have a Lefschetz fibration ## f: W^4 \rightarrow S^2 ## , where ## W^4## is a closed, oriented manifold, and
## S^2## is the 2-disk. This f is a smooth map with finitely-many singularities {## x_1, x_2,..,x_n ##}, so that we have special complex charts ## f(z,w) = z^2+ w^2 ## about each of the critical points ## x_i ##, and we
know the monodromy about each of these critical points; here the monodromy is just an informal measure of how this map differs from being a trivial bundle ## W^4 = S^2 \times F ##, for ##F## the fiber, i.e., the monodromy describes the twisting . And we know that the monodromy for these fibrations is described in terms of Dehn twists about vanishing cycles in the critical fibers. Outside of these singularities, f is just a standard surface bundle.

Question: I have some results for the above bundle, and I want to see which of these results extend/remain when I restrict the base ## S^2 ## to a disk ## D^2## . I am thinking of using the pullback bundle by the restriction map; but I guess we need this ## D^2 ## to include all the critical points. Is there something else I should consider?
Thanks.

EDIT: I also have been told that the charts ## f(z,w)= z^2 + w^2 ## are equivalent to having
charts ## f(z,w) = zw ## . What types of equivalence are we using here; are we saying the
two are equivalent as quadratic forms?
 
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mathwonk

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the equivalence is simply that one can change coordinates to u = z+iw and v = z-iw.

I am not an expert, but will try to say something elementary: when examining the monodromy in your situation, yes one should choose a disc including all singular points, then ordinarily one chooses also another point in the disc which has no singularities over it, and considers arcs that go from that point out to each of the non regular points, then goes once around such a point, and back to the good point. that way all the monodromy is measured at the one good point.

an obvious condition then is that composing all these monodromy loops one has a loop that goes around all the points, hence around none of them looked at wrt the missing point at infinity, hence the composition of these monodromy elements should be the identity.

I believe this study is part of or depends on, the theory of braids. One may consult the works of Moishezon on topology of complex surfaces.

pages 66-67 of the following link may be useful:

http://gokovagt.org/proceedings/2009/ggt09-cataloenwajn.pdf
 
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