×

Stein surfaces as open subsets of \(\mathbb C^2\). (English) Zbl 1118.32011

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.

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
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid