Contents Online
Asian Journal of Mathematics
Volume 26 (2022)
Number 5
Topological convexity in complex surfaces
Pages: 709 – 736
DOI: https://dx.doi.org/10.4310/AJM.2022.v26.n5.a6
Author
Abstract
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.
Keywords
pseudoconvex, smoothing, sliced concordance, almost-complex
2010 Mathematics Subject Classification
32T15, 57-xx
Received 25 July 2022
Accepted 17 August 2022
Published 13 April 2023