Download 4-Manifolds and Kirby Calculus (Graduate Studies in by András I. Stipsicz, Robert E. Gompf PDF

By András I. Stipsicz, Robert E. Gompf

The earlier 20 years have introduced explosive development in 4-manifold conception. Many books are presently showing that strategy the subject from viewpoints resembling gauge conception or algebraic geometry. This quantity, notwithstanding, deals an exposition from a topological standpoint. It bridges the distance to different disciplines and provides classical yet vital topological innovations 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 provides 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 uncomplicated and complete. half III bargains intensive a large diversity of issues from present 4-manifold examine. issues 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 various workouts with suggestions within the ebook.

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

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The union of an increasing sequence of contractible branches is either a contractible branch or the Farey tree. Hence, as a standard application of Zorn’s lemma, one concludes that either S ∼ = Ft or any contractible branch in S is contained in a unique maximal one. If two contractible branches have a common regular vertex, then one of them is contained in the other. Thus, any two distinct maximal contractible branches are disjoint with the possible exception of a common monovalent (for both branches) vertex.

There is a canonical strict deformation retraction Supp◦ S → |S|; 3. the complement Supp S |S| is a union of open disks, one disk for each region. Proof. 2. Hence, Supp S is a locally Euclidean space and the complex orientations of the geometric realizations of all regions match to define an orientation of Supp S. If edges e and e = e ↑ (xy)r are in the same region, the pairs (|reg e|, ψe ) and (|reg e |, ψe ) are canonically homeomorphic: if the width n := wd(reg e) is finite, the homeomorphism is the rotation through 2πr/n about the origin; if n = ∞, it is the translation by 2r along the real axis.

A morphism ϕ : E1 → E2 of bipartite ribbon graphs Si , i = 1, 2, is said to be unramified at a vertex v or region R or S1 if the corresponding ramification index equals one. If all ramification indices are equal to one, then ϕ is said to be unramified; otherwise, it is ramified. Any morphism ϕ : E1 → E2 of graphs extends to a map ϕ˜ : Supp S1 → Supp S2 of their minimal supporting surfaces. To make this extension canonical, for a pair of regions R1 , R2 = ϕ(R1 ) we use the following maps of their geometric realizations: ¯ 2 , z → z r , if wd R1 = r wd R2 < ∞, ¯2 → D • D 2 ¯ →D ¯ , z → exp(2πinz), if wd R1 = ∞ and wd R2 = n < ∞, and • H ¯ → H, ¯ z → z, if wd R1 = wd R2 = ∞.

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