3 Witten ’ s conjecture and Property P

The purpose of this note is to prove the conjecture. The ingredients of the argument are: (a) Taubes’ theorem [11] on the non-vanishing of the SeibergWitten invariants for symplectic 4-manifolds; (b) the theorem of Gabai [7] on the existence of taut foliations on 3-manifolds with non-zero betti number; (c) the construction of Eliashberg and Thurston [4], which produces a contact structure from a foliation; (d) a recent result of Eliashberg [3] on concave filling of contact 3-manifolds; and (e) Witten’s conjecture relating the Seiberg-Witten and Donaldson invariants of smooth 4-manifolds. Although the full version of Witten’s conjecture remains open, a weaker version that is still strong enough to serve our purposes has recently been announced as a theorem by Feehan and Leness [6], following a program proposed by Pidstrigatch and Tyurin. With these ingredients, we shall prove: