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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
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Smooth Four-Manifolds and Complex Surfaces (Paperback, Softcover reprint of hardcover 1st ed. 1994)
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Smooth Four-Manifolds and Complex Surfaces (Paperback, Softcover reprint of hardcover 1st ed. 1994)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 27
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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.
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