By András I. Stipsicz, Robert E. Gompf

ISBN-10: 0821809946

ISBN-13: 9780821809945

The previous 20 years have introduced explosive progress in 4-manifold thought. Many books are presently showing that technique the subject from viewpoints reminiscent of gauge idea or algebraic geometry. This quantity, besides the fact that, bargains an exposition from a topological standpoint. It bridges the space to different disciplines and provides classical yet very important topological recommendations that experience now not formerly seemed within the literature. half I of the textual content provides the fundamentals of the idea on the second-year graduate point and gives an outline of present examine. half II is dedicated to an exposition of Kirby calculus, or handlebody concept on 4-manifolds. it truly is either straightforward and entire. half III bargains intensive a huge diversity of subject matters from present 4-manifold learn. themes contain branched coverings and the geography of advanced surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. functions are featured, and there are over three hundred illustrations and diverse routines with options within the ebook.

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

**Example text**

1) tail(S) is a strong limit bisimulation if and only if for any (P, {Qn: nED}) E S and for any a E Act , (i) if P ~ t», then there exist {Q~: nED} E PN and no ED such that o; ~ Q~ for all n 2: no and (PI, {Q~: nED}) E tail(S); and (ii) if C is a cofinal subset of D and Qm ~ Q~ for all m E C, then there exist P' E P and a cofinal subset B of C such that P ~ P' and (PI , {Q~ : k E B}) E tail(S) . (2) If S is a strong limit bisimulation, then tail(S) is also a strong limit bisimulation. Proof. (2) is immediate from (1) .

O for each nEw, and let a i= b. Then the sequence {Pn : nEw} is not convergent. In fact, if it is not 3. , P rv lim n--+ oo Pn for some PE p , then t here must be P' E P with P ~ P' because PZn ~ 0 for every nE w and {2n: nE w} is a cofinal subset of w, and it follows that there is no E w with Pn ~ for all n ;::: no. O ~ . By genera lizing the preced ing two exa mples, we have t he following example. 6. Let {Pn : nED} be a net of agent s. We write H(Pn ) for the set of imm ediat e act ions of Pn for every n E D.

Indeed, this proposition may be seen as an equational version of the combination of the transition rules for Composition, Restriction and Relabeling. 3. (The Expansion Law) (Pdhll ... (Pdh] I ... I P[[Ji] I .. · I Pn[fn])\L: 1 ~ i ~ n, Pi ~ PI, fi(a) rf. (Pdfd I ... I PflJi] I ... I Pi[fj] I ... I Pn[Jn])\L : 1 ~ i < j ~ n, Pi ~ PI, r, 2.... Pi , fi(li) = h(lj)} ' Proof. pl : P ~ P' ,a E Act and pi E pl. 1) P\L ~ Q if and only if P ~ pi for some pi a rf. 2) P[fl ~ Q if and only if P ~ P' for some (3 E Act with a and for some pi E P with Q == pI [f].

### 4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics, Volume 20) by András I. Stipsicz, Robert E. Gompf

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