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

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

Fiber connected, indefinite Morse 2-functions on connected n-manifolds

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite". This is foundational work for the long-range goal of defining smooth invariants from Morse 2-functions using tools analogous to classical Morse homology and Cerf theory.Comment: 12 pages, 8 figures, to appear in Proc. Nat. Acad. Sci. U.S.A. This contains a condensed presentation of most of the results in arXiv:1102.0750 as well as some alternative perspectives. Revised to add a referenc

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