Download 4-Manifolds and Kirby Calculus by Andras I. Stipsicz Robert E. Gompf PDF

By Andras I. Stipsicz Robert E. Gompf

ISBN-10: 0821809946

ISBN-13: 9780821809945

The prior 20 years have introduced explosive progress in 4-manifold thought. Many books are at the moment showing that procedure the subject from viewpoints akin to gauge idea or algebraic geometry. This quantity, besides the fact that, bargains an exposition from a topological viewpoint. 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 provides the fundamentals of the speculation on the second-year graduate point and gives an outline of present learn. half II is dedicated to an exposition of Kirby calculus, or handlebody idea on 4-manifolds. it truly is either undemanding and finished. half III bargains extensive a large variety of subject matters from present 4-manifold study. issues comprise branched coverings and the geography of advanced surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. purposes are featured, and there are over three hundred illustrations and diverse workouts with options within the e-book.

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Example text

E. We can construct many interesting measures, especially probability distributions, with the next simple result. 19 A → A f dμ is a measure if f ≥ 0 is in L1 . Proof Let ν(A) = A f dμ. To prove ν( i Ei ) = i ν(Ei ) for pairwise disjoint Ei , consider the sequence gn = f 1 ni=1 Ei and note that ∞ . 2 The vector space L1 of integrable functions 39 so the Monotone Convergence Theorem (MCT) completes the proof. e. convergent sequence which is dominated by an integrable function. Fatou’s Lemma holds the key to the proof.

Xn we have 28 Measurable functions/random variables n P (X1 ∈ B1 , Xi ≤ xi , i = 2, . . , n) = P (X1 ∈ B1 ) FXi (xi ). i=2 Then C1 contains every interval of the form (−∞, x] so it contains the π-system that generates B. If we can show that C1 is a d-system, the Monotone Class Theorem implies that our identity holds for all Borel sets B1 on the line. Assume for a moment that this has been done. We can then iterate the procedure for each index: for example, we let C2 be the class of Borel sets B2 for which P (X1 ∈ B1 , X2 ∈ B2 , Xi ≤ xi , i = 3, .

P ): this is an equivalence relation on Lp (Ω) and we denote the quotient space Lp (Ω)/ ∼ by Lp (Ω). e. we identify an equivalence class with any of its representatives. To see that (Lp (Ω), ||·||) is a normed vector space, note first that if X, Y are random variables, then so are X + Y, cX for real c. Since 49 Lp -spaces and conditional expectation 50 p p p p p p |cX| = |c| |X| and |X + Y | ≤ 2p max{|X| , |Y | }, it follows that Lp (Ω) is a vector space and ||cX||p = |c| |X||p . ). So all that remains is to prove the triangle inequality for this norm.

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