The notion of the Hausdorffized leaf space of a foliation is introduced. A sufficient condition for warped compact foliations to converge to is given. Moreover, a necessary condition for warped compact Hausdorff foliations to converge to is shown. Finally, some examples are examined.
We prove several results on weak symplectic fillings of contact -manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable—this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori—this gives many...
We prove that Witten’s Conjecture [40] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with and odd follows from our -monopole cobordism formula [6] when the four-manifold has or is abundant.