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

**Example text**

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.