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 idea. Many books are at the moment showing that process the subject from viewpoints corresponding to gauge idea or algebraic geometry. This quantity, although, deals an exposition from a topological perspective. It bridges the space to different disciplines and provides classical yet very important topological concepts 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 idea on 4-manifolds. it's either easy and entire. half III deals intensive a huge diversity of themes from present 4-manifold learn. subject matters comprise 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 routines with suggestions within the booklet.

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It is easy Hence, Nf is a T-set in 1 2Tn and Mf is the only component of Zn — M . Since M t +i C Mt- for each i > 713, it follows that N% C JV£+1 C .... Let Nn = c l ( U £ n ^ f )• Claim 3. 1, Nn is a T-set in Zn. If n > m > n3 then h^(Nn) Proof of Claim 3. = 7Vm. _ Since the sequences (Nf) suffices to prove that h^Nf) = Nf 1 for each j are increasing, it >n. Let j > n. *) is a T-set in Zm, Lf(x') E K{Zn-B{) € K ( Z m - £ > , ) . (z') and ^ ( ( ^ J - ^ i i C ^ ) ) ) = C O " 1 ^ * ' ) ) C Lf{x'), it follows that Lf (*') PI /»£(£*(*')) # 0.

Obviously, Y is homeomorphic to Y' — lim inv (Yn,gn,n > n0). Thus we may assume that Y = Y'. i > no). rn(p), qn = ^n(q) and r n = 7Tn(r). If n > no an let n™ = idz n5 n" +1 d if n 0 < n < m let /i™ = ftn o . . o n m _i : Z m —* Zn. In particular, = /i n and all the maps ft™ are monotone surjections. Let n > no and a n : /„ -^ Xn be a continuous map of some arc In onto Xn- Then A n — ctn (Zn^ is a closed subset of JTn. Hence, A n is an ordered compactum. Therefore Zn — an(Kn) is a continuous image of an ordered compactum.

Let bd(/\(^)) = {a, 6}. Let # , # ' G -Fm be such that and 6 G / / ' . Then H,H' X By Theorem cl A aeH G ^m — {F, G} and F and G are in distinct components of -(HUH'). 3 is complete. 4. Theorem. Let X be a locally connected continuum with dimX = 1. If X is a continuous image of an arc, then there exists an inverse sequence (Xn,fn) such that each Xn is a rim-finite continuum, and each fn : Xn+\ —• Xn is a monotone surjection and X = lim inv (Xn, Proof. 3, there exists an inverse sequence (Ynjq^) such that Y = lim inv ( Y n , ( ^ ) , each Yn is a rim-finite continuum, and each q% is a monotone surjection.

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