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.

中文翻译：

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复杂曲面中的拓扑凸性

我们在复杂曲面中拓扑（通常不平滑）嵌入 $3$ 流形的背景下研究严格伪凸性的概念。拓扑伪凸 (TPC) $3$-流形与其光滑类似物的行为相似，切掉了全纯的开放域（Stein 曲面），但它们更为常见。我们提供了构建 TPC 嵌入的工具，并表明每个封闭的、有方向的 $3$-流形 $M$ 都有一个 TPC 嵌入到紧凑、复杂的曲面（无边界）中，实现了几乎复杂结构的任何同伦类（类似于光滑情况下接触平面场的同伦类）。我们用不变量证明了我们的工具定理，这些不变量对任何 $4$ 流形同伦等价于 $M$ 上的几乎复杂结构进行分类。这些不变量易于计算并受到同胚（不一定平滑）的尊重。我们研究了带线的 $3$ 流形和带领端的产品的两个等价平滑类。这两类平滑都是通过全纯嵌入来实现的，全纯嵌入展示了几乎复杂结构的任何预先指定的同伦类。一类来自 TPC 嵌入式 $3$ 流形，而另一类可能不是。