#### WWGD

Science Advisor

Gold Member

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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?

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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