By András I. Stipsicz, Robert E. Gompf

The prior 20 years have introduced explosive progress in 4-manifold idea. Many books are at present showing that technique the subject from viewpoints resembling gauge thought or algebraic geometry. This quantity, notwithstanding, bargains an exposition from a topological standpoint. It bridges the space to different disciplines and offers classical yet vital topological ideas 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 outline of present study. half II is dedicated to an exposition of Kirby calculus, or handlebody conception on 4-manifolds. it's either straight forward and complete. half III deals extensive a wide variety of themes from present 4-manifold examine. issues 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 ideas within the booklet.

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

**Example text**

M} is a bijection of the ﬁnite set {1, . . , n} with a proper subset of itself, contradicting the corollary just proved. 6. If B is a subset of the ﬁnite set A, then B is ﬁnite. If B is a proper subset of A, then the cardinality of B is less than the cardinality of A. 7. Let B be a nonempty set. Then the following are equivalent: (1) B is ﬁnite. (2) There is a surjective function from a section of the positive integers onto B . (3) There is an injective function from B into a section of the positive integers.

In the preceding theorem, however, we were not really proving a theorem by induction, but deﬁning something by induction. How then should we start the proof? Can we start by saying, “Let A be the set of all integers n for which the function h is deﬁned”? But that’s silly; the symbol h has no meaning at the outset of the proof. It only takes on meaning in the course of the proof. So something more is needed. What is needed is another principle, which we call the principle of recursive deﬁnition.

3. Let X be the two-element set {0, 1}. Show there is a bijective correspondence between the set P (Z+ ) and the cartesian product X ω . 4. (a) A real number x is said to be algebraic (over the rationals) if it satisﬁes some polynomial equation of positive degree x n + an−1 x n−1 + · · · + a1 x + a0 = 0 with rational coefﬁcients ai . Assuming that each polynomial equation has only ﬁnitely many roots, show that the set of algebraic numbers is countable. (b) A real number is said to be transcendental if it is not algebraic.

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