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Read e-book online 4-Manifolds and Kirby Calculus PDF

By Andras I. Stipsicz Robert E. Gompf

ISBN-10: 0821809946

ISBN-13: 9780821809945

The previous 20 years have introduced explosive progress in 4-manifold conception. Many books are presently showing that strategy the subject from viewpoints akin to gauge conception or algebraic geometry. This quantity, in spite of the fact that, bargains an exposition from a topological perspective. It bridges the space to different disciplines and offers classical yet vital topological recommendations that experience now not formerly seemed within the literature. half I of the textual content offers 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 concept on 4-manifolds. it truly is either undemanding and accomplished. half III bargains extensive a wide diversity of issues from present 4-manifold study. themes comprise branched coverings and the geography of complicated 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 strategies within the e-book.

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Moreover, |d f xi | ≤ M. Therefore, A xi (Bri (x i )) is contained in a (k − 1)-dimensional disk D i ⊂ Vi of radius Mr i . Hence I ε (x i , ri ) ⊂ z ∈ Rn : dist (z, Di ) ≤ εri . 12) Then, it is elementary to check that each I ε (x i , ri ) can be covered by Cε −(k−1) ndimensional balls Bi, j of radius εr i (where the constant C depends only on k, m, and M). 11) we obtain that {B i, j } is a countable covering of F. Moreover, the diameter of each Bi, j is precisely 2εr i ≤ 2ε. 10) ≤ ωk C5k R k ε .

Y there exist a positive constant c y and a k-dimensional linear plane V y such that r −k µ y,r ∗ cy H k Vy as r ↓ 0. The Rectifiability Criterion. We now come to an important question: How does one prove that a set is rectifiable? 6. Before stating it, we introduce some notation. 5 (k-cones). Let V be a k-dimensional linear plane of R n . Then we denote by V ⊥ the orthogonal complement of V . Moreover, we denote by P V and Q V , respectively, the orthogonal projection on V and V ⊥ . For α ∈ R+ , we denote by C(V , α) the set x ∈ Rn : |Q V (x)| ≤ α|PV (x)| .

3) Proof. We define R(E) := E : E ⊂ E is Borel and rectifiable . and α := sup H k (E ) . E ∈R(E) Let {E i } ⊂ R(E) be such that H k (E i ) ↑ α. Then we set E r := E i . Clearly E r is rectifiable, E r ⊂ E, and H k (E r ) = α. We claim that E c := E \ E r is purely unrectifiable. Indeed, if there were a Lipschitz graph Γ such that H k (E c ∩ Γ) > 0 we would have that H k E r ∪ (Γ ∩ E c ) > α . 4) would contradict the maximality of α. To prove the uniqueness of the decomposition, note that the intersection between a purely unrectifiable set and a rectifiable set has always H k measure 0.

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

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