4-manifolds (Q2800601)

Many reasons make the study of PL (or smooth) manifolds in dimension four extremely interesting, as evidenced by the numerous, significant texts that in the last decades have dealt with the subject, from different points of view: for example, [\textit{R. C. Kirby}, The topology of 4-manifolds. Berlin etc.: Springer-Verlag (1989; Zbl 0668.57001), \textit{M. H. Freedman} and \textit{F. S. Quinn}, Topology of 4-manifolds. Princeton, NJ: Princeton University Press (1990; Zbl 0705.57001) and \textit{R. E. Gompf} and \textit{A. I. Stipsicz}, 4-manifolds and Kirby calculus. Providence, RI: American Mathematical Society (1999; Zbl 0933.57020)], only to cite the most famous ones.NEWLINENEWLINEAkbulut's book provides an up-to-date overview of the subject, with a strongly original, intuitive approach. Key constructions and theorems are presented through a discursive flow of arguments, which has the vitality of true lessons, where learners are called to interact actively with the teacher. Indeed, demonstrative arguments often focus on the most meaningful passages, leaving to the reader, through suitably provided exercises, the task of completing the details not covered by the proofs. The author also performs a precise selection among the subjects, sometimes emphasizing particular cases, when their understanding is thought to be ``instructive'' in order to understand many of the difficulties appearing in the general case. In addition, most of the proofs are made through drawings, which are the real peculiarity of the text. The author himself explains this choice in the Preface: ``When there was a chance to explain some idea with pictures I always went with pictures, staying true to the saying \textit{A picture is worth a thousand words}''.NEWLINENEWLINEAlthough it is clearly stated that there is ``no attempt to cover all related results around the materials discussed'', the book succeeds in presenting a wide collection of the most significant recent results on the topology of smooth 4-manifolds: ``4-manifold handlebodies'' is both the name of the starting Chapter and -- due to the choice to make illustrations of most proofs -- the working tool all along the book; other chapters deal -- for example -- with ``Building low-dimensional manifolds'', ``Gluing 4-manifolds along their boundaries'', ``Bundles'', ``Lefschetz fibrations'', ``Symplectic manifolds'', ``Exotic 4-manifolds'', ``Cork decomposition'', ``Complex surfaces'', ``Seiberg-Witten invariants''.NEWLINENEWLINEThe final chapter (``Some applications'') is the climax of the whole book: indeed, the techniques developed in the earlier chapters are applied to construct -- via sequences of beautiful figures, supplemented by very few words -- handlebodies of some closed symplectic manifolds which are exotic copies of the standard manifolds \(\mathbb{C P}^2 \# 3 \bar{\mathbb{C P}}^2\) and \(\mathbb{ C P}^2 \# 2 \bar{\mathbb{ C P}}^2\), respectively. These exotic manifolds have been originally built by \textit{A. Akhmedov} and \textit{B. D. Park} [Invent. Math. 173, No. 1, 209--223 (2008; Zbl 1144.57026); ibid. 181, No. 3, 577--603 (2010; Zbl 1206.57029)], while the presented constructions have been obtained by the author himself in [\textit{S. Akbulut}, Adv. Math. 274, 928--947 (2015; Zbl 1384.57020); ibid. 274, 948--967 (2015; Zbl 1421.57042)].NEWLINENEWLINEReally a fascinating book, both for researchers and for graduate students interested in the subject.