By András I. Stipsicz, Robert E. Gompf

The prior 20 years have introduced explosive progress in 4-manifold concept. Many books are presently showing that procedure the subject from viewpoints similar to gauge concept or algebraic geometry. This quantity, despite the fact that, bargains an exposition from a topological viewpoint. It bridges the space to different disciplines and provides classical yet vital topological strategies that experience no longer formerly seemed within the literature. half I of the textual content provides the fundamentals of the idea on the second-year graduate point and provides an outline of present learn. half II is dedicated to an exposition of Kirby calculus, or handlebody idea on 4-manifolds. it really is either uncomplicated and accomplished. half III bargains extensive a vast diversity of issues from present 4-manifold learn. subject matters comprise branched coverings and the geography of complicated 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 workouts with suggestions within the booklet.

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

**Sample text**

Let us show that there exists a Moscow group G such that X x G is not a PT-group. In fact, G can be selected to be a topological group of countable pseudocharacter. 7 are closed, and the closures of elements of r/in # X do not cover # X . Note, that # X is, in this case, the product of w~ copies of D = {0, 1}. 7. Clearly, G is a topological group. It is also obvious that G is Ra~ov complete. 7 such that U7 is dense in X. For each P E "7 and each positive n E w, the set Up, n of all f E G such that If(z)] < 1In for every z E P is open in G and contains the zerofunction 0 on X which is the neutral element of G.

In this case, the Bohr topology on G coincides with the weak topology generated by the family of all continuous characters on G, and G with this topology is denoted by G +. In particular, all locally compact Abelian groups belong to the class of MAP groups. The Bohr topology of locally compact Abelian groups was studied in many articles, some of them quite recent. 3. THEOREM.

The topological group (7* so obtained is a P-space. Then G* is not a Moscow space. Therefore, G* may be a candidate for being a non-PT-group. However, the solution to the PT-problem in ARHANGEL'SKII [2000a] presented below is based on a different idea. A topological group (7 is called Ro-bounded provided that for each neighborhood V of the neutral element there exists a countable subset A of (7 such that A V = G. Tka~enko has shown that every R-factorizable group is Ro-bounded and that not every lqo-bounded group is R-factorizable (see TKAt~ENKO [1991b]).