3 0 N ov 2 00 3 Witten ’ s conjecture and Property
3 0 N ov 2 00 3 Witten ’ s conjecture and Property
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3 0 N ov 2 00 3 维滕猜想及其性质
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
T. Mrowka
中科院分区：
文献类型：
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作者：
P. Kronheimer;T. Mrowka
这篇笔记的目的是为了证明这个猜想。这一论点的主要内容是：(A)关于辛4-流形的SeibergWitten不变量不消失的Taubes定理[11]；(B)Gabai[7]关于具有非零Betti数的3-流形上的张叶的存在性的定理；(C)由叶层构造出一个接触结构；(D)Eliashberg[3]关于接触3-流形的凹填充的最新结果；以及(E)Witten关于光滑4-流形的Seiberg-Witten和Donaldson不变量的猜想。尽管Witten猜想的完整版本仍然开放，但在Pidstrigatch和Tyurin提出的计划之后，Feehan和Lness[6]最近已经建立了一个更弱的版本，它仍然足够强大，足以服务于我们的目的。有了这些要素，我们将证明：
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 established by Feehan and Leness [6], following a program proposed by Pidstrigatch and Tyurin. With these ingredients, we shall prove: