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)
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 = ∞.