# Smooth Four-Manifolds and Complex Surfaces

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 27)

Advertisement

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 27)

In 1961 Smale established the generalized Poincare Conjecture in dimensions greater than or equal to 5 [129] and proceeded to prove the h-cobordism theorem [130]. This result inaugurated a major effort to classify all possible smooth and topological structures on manifolds of dimension at least 5. By the mid 1970's the main outlines of this theory were complete, and explicit answers (especially concerning simply connected manifolds) as well as general qualitative results had been obtained. As an example of such a qualitative result, a closed, simply connected manifold of dimension 2: 5 is determined up to finitely many diffeomorphism possibilities by its homotopy type and its Pontrjagin classes. There are similar results for self-diffeomorphisms, which, at least in the simply connected case, say that the group of self-diffeomorphisms of a closed manifold M of dimension at least 5 is commensurate with an arithmetic subgroup of the linear algebraic group of all automorphisms of its so-called rational minimal model which preserve the Pontrjagin classes [131]. Once the high dimensional theory was in good shape, attention shifted to the remaining, and seemingly exceptional, dimensions 3 and 4. The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver. Thus new ideas are necessary to study manifolds of these "low" dimensions.

4-Manifold 4-Mannigfaltigkeiten Anti-Self-Dual Connection Complex Surface Donaldson Polynomial Donaldson'sches Polynom Elliptic Surface Elliptische Fläche Four-Manifold Gauge Theory Komplexe Fläche algebraic geometry diffeomorphism manifold topology

- DOI https://doi.org/10.1007/978-3-662-03028-8
- Copyright Information Springer-Verlag Berlin Heidelberg 1994
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-642-08171-2
- Online ISBN 978-3-662-03028-8
- Series Print ISSN 0071-1136
- Buy this book on publisher's site