Reconstructing 4-manifolds from Morse 2-functions

Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions.Comment: 13 pages, 10 figures. Replaced because the main theorem in the original is false. The theorem has been corrected and counterexamples to the original statement are give

Constructing symplectic forms on 4-manifolds which vanish on circles

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper20.abs.htm

Indefinite Morse 2-functions; broken fibrations and generalizations

A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions mapping to arbitrary compact, oriented surfaces. "Uniqueness" means there is a set of moves which are sufficient to go between two homotopic indefinite Morse 2-functions while remaining indefinite throughout. We extend the existence and uniqueness results to indefinite, Morse 2-functions with connected fibers.Comment: 74 pages, 41 figures; further errors corrected, some exposition added, other exposition improved, following referee's comment

Quasi-planar steep water waves

A new description for highly nonlinear potential water waves is suggested, where weak 3D effects are included as small corrections to exact 2D equations written in conformal variables. Contrary to the traditional approach, a small parameter in this theory is not the surface slope, but it is the ratio of a typical wave length to a large transversal scale along the second horizontal coordinate. A first-order correction for the Hamiltonian functional is calculated, and the corresponding equations of motion are derived for steep water waves over an arbitrary inhomogeneous quasi-1D bottom profile.Comment: revtex4, 4 pages, no figure