4-Manifolds and Kirby Calculus (Graduate Studies in by András I. Stipsicz, Robert E. Gompf

By András I. Stipsicz, Robert E. Gompf

The prior twenty years have introduced explosive progress in 4-manifold idea. Many books are at the moment showing that strategy the subject from viewpoints akin to gauge idea or algebraic geometry. This quantity, even if, deals an exposition from a topological standpoint. It bridges the distance to different disciplines and provides classical yet vital topological suggestions that experience no longer formerly seemed within the literature. half I of the textual content offers the fundamentals of the idea on the second-year graduate point and gives an outline of present examine. half II is dedicated to an exposition of Kirby calculus, or handlebody concept on 4-manifolds. it truly is either easy and accomplished. half III deals extensive a wide variety of subject matters from present 4-manifold examine. subject matters comprise branched coverings and the geography of advanced surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. functions are featured, and there are over three hundred illustrations and diverse workouts with options within the publication.

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Then there exists an automorphism T;, Kakutani equivalent to T, and having a factor-automorphism which is metrically isomorphic to T,. 5 (cf [ORW]). If T,, T, are FF-automorphisms and both are either of zero entropy, or of positive entroppi, then they are Kakutani equivalent. The analogy between the ‘‘$theory” which was sketched above and the Ornstein “&theory” turns out to be so complete that, unfortunately, beyond the class of LB-automorphisms (just as in the Ornstein theory-beyond the class of B-automorphisms) the results are mostly negative.

A::)) the vector of the lengths of intervals exchanged by T", n = 1, 2, . . '. 2. An interval exchange transformation has Property LY iffor some > 0 the set A = A(T,&)= { n E N : m,(T) 2 &/n}is essential. 5 (M. Boshernitzan). Let T be a minimal interval exchange transformation which satisfies Property 9. Then T is uniquely ergodic. ndoes not is of zero Lebesgue measure. satisfy Property 9, Further results on the ergodic and spectral properties of "typical" interval exchange transformations were obtained by W.

G. Sinai 54 Chapter 3. Entropy Theory of Dynamical Systems 55 2 ) thenatural valuedfunctionsf, E L ' ( M , , A l , p I ) , f 2 E L ' ( M 2 , ~ , , p 2 ) e x i s t such that the integral automorphisms (T,)Il,( T2)fzare metrically isomorphic. 1. The above relation is transitive, so it is really an equivalence relation. S. Kakutani introduced this relation in connection with his theorem on special representations of flows. The purpose was to describe the class of all possible special representations for a given flow.

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