University of Toronto's Symplectic Geometry Seminar

Dec 1, 2003, 2:00-2:50 PM

David Gay


"Harmonic 2-forms on 4-manifolds by way of handles"


(Joint with R. Kirby.) Given a handlebody description of a closed 4-manifold X, with b_2^+(X) > 0, we will show how to build a harmonic 2-form on X representing a given cohomology class. This form is 0 along a collection of circles and symplectic everywhere else, with a compatible almost complex structure away from the circles. The construction can be tailored to realize certain given spin^c structures. The point is to feed this construction into a program of Taubes to extract topological invariants by counting J-holomorphic curves with respect to such a 2-form.