Download e-book for iPad: 4-Manifolds and Kirby Calculus (Graduate Studies in by András I. Stipsicz, Robert E. Gompf

By András I. Stipsicz, Robert E. Gompf

ISBN-10: 0821809946

ISBN-13: 9780821809945

The earlier twenty years have introduced explosive progress in 4-manifold idea. Many books are at present showing that procedure the subject from viewpoints resembling gauge thought or algebraic geometry. This quantity, despite the fact that, deals an exposition from a topological perspective. It bridges the distance to different disciplines and provides classical yet very important topological options that experience no longer formerly seemed within the literature. half I of the textual content provides the fundamentals of the speculation on the second-year graduate point and gives an summary of present examine. half II is dedicated to an exposition of Kirby calculus, or handlebody conception on 4-manifolds. it really is either straightforward and finished. half III bargains intensive a vast variety of subject matters from present 4-manifold learn. issues contain 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 various workouts with ideas within the e-book.

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Extra info for 4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics, Volume 20)

Example text

The fact stated below, concerning standard probability spaces, will be of crucial importance in many arguments throughout the book. We refer the reader to [Rokhlin, 1952] for the proof and for more background on standard spaces. 2 Let (X, A, μ) be a standard probability space. , ν(B) = μ(π −1 (B)) (B ∈ B ). We will write ν = πμ. , it is, up to measure, a bimeasurable bijection). By atoms of B we will understand the preimages π −1 (y) of points y ∈ Y . The above fact implies that any A-measurable set which is a union of atoms of B, is B-measurable.

Applying this to the probability 30 Shannon information and entropy vector p = pi ni i pi ni i≤m and by convexity of the function − log t, we obtain max Ip ,n (i) + max n1i ≥ H(p , n) = i i 1 i pi ni − pi log pi − i pi log ni + log i pi ni i ≥ H(p) . 14). 15) note that there is c ≤ c such that p = (2−c ni )i is a probability vector and then Ip ,n (i) = c for each i. 16) yields H(p, n) ≤ c + maxi n1i ≤ c + maxi n1i for any finite-dimensional p. Approximating an arbitrary probability vector p by the finite-dimensional vectors p(m) and because m−1 H(p, n) = sup − m i=1 log pi ni ≤ sup H(p(m) , n), m we extend the inequality to all probability vectors.

K} and we calculate the Shannon entropy of the join PF . Because Q is refined by each Pi , it is also refined by PF , thus we have H(PF ) = H(PF ∨ Q) = H(PF |Q) + H(Q) = μ (B)HB (PF ) + μ (A)HA (PF ) + H(Q) = n−1 n · 0 + n1 Hμ (PF ) + H(Q) = 1 n Hμ (PF ) + H(Q). The error term H(Q) of this approximation depends only on n and converges to zero as n → ∞. This concludes the proof. The closure Γk remains hard to describe; only for k ≤ 3 it is determined by the Shannon inequalities. , it cannot be described by a system of linear inequalities.

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4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics, Volume 20) by András I. Stipsicz, Robert E. Gompf

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