Gompf, Robert E. Stein surfaces as open subsets of \(\mathbb C^2\). (English) Zbl 1118.32011 J. Symplectic Geom. 3, No. 4, 565-587 (2005). In this interesting paper, the author discusses several topics concerning the Stein manifold structures on 4-manifolds by using the language of the handlebody theory. Thus, he shows that an open subset \(U\) of a complex surface \(X\) can be topologically perturbed to yield an open subset of \(X\) whose inherited complex structure is Stein iff \(U\) is homeomorphic to the interior of a handlebody whose handles all have index \(\leq 2\). However, the differential topology is altered and frequently there are uncountably many possibilities for the resulting diffeomorphism type of \(U\). Several results which will be presented in a forthcoming paper are also described. Reviewer: Eugen Pascu (Montréal) Cited in 1 ReviewCited in 9 Documents MSC: 32E10 Stein spaces 32Q28 Stein manifolds 57M50 General geometric structures on low-dimensional manifolds 57N16 Geometric structures on manifolds of high or arbitrary dimension 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 32Q35 Complex manifolds as subdomains of Euclidean space 57R65 Surgery and handlebodies 57R60 Homotopy spheres, Poincaré conjecture Keywords:Stein surface; 4-manifolds; surgery and handlebodies; exotic spheres; Morse theory × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid