Smooth Four-Manifolds and Complex Surfaces
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Beschreibung
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.
With this book, appearing in Springer's prestigious series Ergebnisse der Mathematik, Friedman and Morgan give a comprehensive introduction to current research in the field of applications of gauge theory to complex surfaces. Their streamlined modern presentation of this very active area of mathematical research will be very welcome to researchers and graduate students alike. The study is divided into 4 main areas: - Classical complex surface theory; - gauge theory and Donaldson invariants; - deformations of holomorphic vector bundles; - explicit calculations for elliptic surfaces. The book very successfully bridges the two areas of 4-manifold topology and complex surface theory, thus unifying the classical theory of Barth/Peters/van de Ven (ERGEBNISSE 4) with gauge theory techniques described by Donaldson/Kronheimer.
Autor*in
Robert Friedman
Themen in »Smooth Four-Manifolds and Complex Surfaces«
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