By Andras I. Stipsicz Robert E. Gompf

The previous 20 years have introduced explosive development in 4-manifold concept. Many books are presently showing that technique the subject from viewpoints resembling gauge concept or algebraic geometry. This quantity, even if, bargains an exposition from a topological viewpoint. It bridges the distance to different disciplines and provides classical yet vital topological thoughts that experience now not 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 summary of present examine. half II is dedicated to an exposition of Kirby calculus, or handlebody thought on 4-manifolds. it truly is either trouble-free and complete. half III deals extensive a huge variety of issues from present 4-manifold examine. subject matters contain 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 recommendations within the e-book.

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**4-Manifolds and Kirby Calculus**

The previous twenty years have introduced explosive development in 4-manifold idea. Many books are at the moment showing that strategy the subject from viewpoints corresponding to gauge idea or algebraic geometry. This quantity, despite the fact that, bargains an exposition from a topological perspective. It bridges the distance to different disciplines and offers classical yet vital topological recommendations that experience no longer formerly seemed within the literature.

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**Extra resources for 4-Manifolds and Kirby Calculus**

**Example text**

V1–V8 are all immediate consequences of our definitions and the properties of . For example, to prove V6, let x = (x1, . . , xn). Then The remaining verifications are left as exercises for the student. A vector space is a set V together with two mappings V × V → V and × V → V, called vector addition and scalar multiplication respectively, such that V1–V8 above hold for all and (V3 asserts that there exists such that x + 0 = x for all , and V4 that, given , there exists such that x + (− x) = 0).

Vn for V. Since it is clear (by linearity of L) that the vectors L( v1), . . , L( vn) generate Im L, it suffices to prove that they are linearly independent. Suppose Then so t1 v1 + · · · + tn vn = 0 because Ker L = 0. But then t1 = · · · = tn = 0 because the vectors v1, . . , vn are linearly independent. 3). 2 Let L : n → m be defined by L( x) = A x, where A = (aij) is an m × n matrix. Then (a) Ker L is the orthogonal complement of that subspace of n that is generated by the row vectors A1, .

3). 2 Let L : n → m be defined by L( x) = A x, where A = (aij) is an m × n matrix. Then (a) Ker L is the orthogonal complement of that subspace of n that is generated by the row vectors A1, . . , Am of A, and (b) Im L is the subspace of m that is generated by the column vectors A1, . . , An of A. PROOF (a) follows immediately from the fact that L is described by the scalar equations so that the ith coordinate Li( x) is zero if and only if x is orthogonal to the row vector Ai. (b) follows immediately from the fact that Im L is generated by the images L( e1), .