## Topology of 4-manifolds.(English)Zbl 0705.57001

Princeton Mathematical Series, 39. Princeton, NJ: Princeton University Press. viii, 259 p. \$ 49.50 (1990).
Topology of 4-manifolds is widely recognized as one of the presently most exciting areas of mathematics, having its very own unique flavour and complication. The authors, both very distinguished specialists in this field, have jointly written a very attractive and most comprehensive survey of the current state of the art.
There are two major parts in this book: the embedding theorems and their applications to the study of topological 4-manifolds. The first part starts with the basic tools for manipulation of maps of surfaces into 4- manifolds, in particular Whitney moves and transverse spheres. It continues with capped gropes and towers, the principal technical innovation discovered since the original publications on the subject. Next, a tribute is paid to the classical masters of the decomposition space techniques, R. H. Bing and M. Brown. The first part culminates in the proof of the embedding theorem and its variants.
Part two develops several consequences of the embedding theorems which are then demonstrated to be most useful in the study of structure of topological 4-manifolds. More precisely, in the beginning the basic machinery, previously known to be available in other dimensions, is developed, in particular variations on the 5-dimensional h-cobordism theorem, smooth structures on 4-manifolds, handlebody structures, normal bundles, and transverse approximations. Then specific structural results are given: the classification theorem and surgery for 4-manifolds with several interesting consequences, in particular for knots in the 3-sphere and 4-sphere. In the conclusion, attempts are described to extend the class of fundamental groups for which the main result, embedding theorem, is still valid and some straightforward reformulations in terms of slices for certain families of links, and less direct ones involving transversality of Poincaré spaces, and actions of free groups on the 3- sphere are given.
There are many exercises scattered through the book and the reader will also find good historical references. The book is accessible for most graduate students. I enthusiastically recommend it to a much wider audience: to everyone who whishes to get familiar with the mysterious world of 4-dimensions and then try his hands at some of the numerous unresolved problems to which the authors skilfully direct us.
Reviewer: D.Repovš

### MSC:

 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57R40 Embeddings in differential topology 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 57R80 $$h$$- and $$s$$-cobordism 57R65 Surgery and handlebodies