论文标题
复杂表面的拓扑凸度
Topological convexity in complex surfaces
论文作者
论文摘要
我们研究了严格的假发性在拓扑（通常不平滑）嵌入复杂表面中的3个manifolds的概念。拓扑假数（TPC）3个manifolds的行为与它们的平滑类似物相似，从而削减了Holomorphy（Stein表面）的开放域，但它们更为普遍。我们提供用于构建TPC嵌入的工具，并表明每个封闭的，定向的3个manifold M都在紧凑的，复杂的表面（无边界）中嵌入一个TPC，以实现任何几乎复合结构的同型类别（平滑情况下接触平面场的同质类似物的类似物）。我们证明了我们的工具定理具有不变性，这些工具将几乎复杂的结构分类为等同于M的任何4个manifold同型。这些不变性可以适合计算并受同构形态（不一定平滑）尊重。我们研究了两个等效类别的平滑等等类，这些平滑线在3个manifold的乘积上，以及一条线的末端。两类的平滑条件都是通过显示出任何签名的几乎复合结构的同型同型类别的塑形嵌入来实现的。一个类是由TPC嵌入的3个manifolds产生的，而另一个可能不是。
We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented 3-manifold M has a TPC embedding in a compact, complex surface (without boundary) realizing any homotopy class of almost-complex structures (the analogue of the homotopy class of the contact plane field in the smooth case). We prove our tool theorems with invariants that classify almost-complex structures on any 4-manifold homotopy equivalent to M. These invariants are amenable to computation and respected by homeomorphisms (not necessarily smooth). We study the two equivalence classes of smoothings on the product of a 3-manifold with a line, and on collared ends. Both classes of smoothings are realized by holomorphic embeddings exhibiting any preassigned homotopy class of almost-complex structures. One class arises from TPC embedded 3-manifolds, while the other likely does not.
