[A complete version of this abridged review is available on request.]
In classical commutative algebra the desire to simplify certain calculations by concentrating on a certain subset of primes P and to allow division by the primes outside P led to the concept of localisation. The theory of the localisation of (non-abelian) groups appears to have been first studied by A. I. Malcev [Izv. Akad. Nauk SSSR, Ser. Mat. 13, 201-212 (1949; Zbl 0034.017)] in the case of torsionfree nilpotent groups; this approach was reworked and extended by M. Lazard [Ann. Sci. Éc. Norm. Supér., III. Sér. 71, 101-190 (1954; Zbl 0055.251)], and others [G. Baumslag, Proc. Am. Math. Soc. 12, 262-267 (1961; Zbl 0101.262); Lecture Notes on Nilpotent groups, Reg. Conf. Ser. Math. 2 (1971; Zbl 0241.20001); the author, Math. Z. 132, 263-286 (1973; Zbl 0264.20037); D. Quillen, Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.537); R. B. Warfield jun., Nilpotent groups, Lect. Notes Math. 513 (1976; Zbl 0347.20018)]. Localisation in homotopy theory seems to have been used first by J. F. Adams [Q. J. Math., Oxf. II. Ser. 12, 52-60 (1961; Zbl 0119.187)] in studying multiplications on spheres away from 2. One of the early peaks of the theory was the discovery, due to the author and J. Roitberg, of a non-classical finite H-space [Ann. Math., II. Ser. 90, 91-107 (1969; Zbl 0159.539)]; here a classical H-space is one whose homotopy type is that of a product of a Lie group with a number of seven spheres.
Localisation then became a standard tool in homotopy theory. Among others, its development was prompted by D. Sullivan [Geometric topology, part I: Localization, periodicity, and Galois symmetry. M.I.T. Press, Cambridge, Ma. (1970; for a review of the Russian translation (1975) see Zbl 0366.57003); Ann. Math., II. Ser. 100, 1-79 (1974; Zbl 0355.57007)], A. K. Bousfield and D. M. Kan [Homotopy limits, completions and localizations, Lect. Notes Math. 304 (1972; Zbl 0259.55004)], and by the author, G. Mislin and J. Roitberg [Localization of nilpotent groups and spaces (1975; Zbl 0323.55016)]. Among the numerous results obtained with the aid of localisation in homotopy theory we only cite those of F. R. Cohen, J. C. Moore and J. A. Neisendorfer [Ann. Math., II. Ser. 109, 121-168 (1979; Zbl 0405.55018); ibid. 110, 549-565 (1979; Zbl 0443.55009)] and those of P. Selick [Thesis, Princeton University, 1977] on torsion in homotopy groups.
The book under review (and the cited monograph of Hilton, Mislin and Roitberg also) proceeds by making a purely algebraic study of the group- theoretical aspects of the localisation method; thereby the inessential topology somewhat has been stripped away, and only standard homological algebra together with elementary group theory is used.
The book is written in German. It is, in the author’s words, ”a faithful record of a year long course given as a Nachdiplomvorlesung” (i.e. graduate course) ”at the ETH Zürich, during the academic year 1981/82”. Due to the circumstances of the course, ”it was necessary to start at an elementary level and to be very detailed in the treatment”. The prospective reader should note the words ”elementary” and ”very detailed in the treatment” in that sentence. Only the elementary parts of the subject are discussed and, while ”the content of these notes overlaps very considerably with that of the monograph by Hilton, Mislin and Roitberg [loc. cit.], the style and manner of treatment are very different”. Indeed, in the monograph, the primary concern is really with the localisation of nilpotent homotopy types, and so only those aspects of the theory of localisation of nilpotent groups are discussed in detail which are relevant to homotopy theory. This somewhat contrasts with the book under review, in which the treatment of localisation of nilpotent groups is more detailed, and which also contains, in Section 5, a number of new algebraic results; on the other hand, the part on localisation of homotopy types contains much less information than the monograph. Sections 1-3 of the book constitute a self-contained introduction to the theory of nilpotent groups, and Section 4 explains the theory of localisation of abelian groups. In Section 5 a localisation theory for nilpotent groups is given. There follow in Section 5 the ”pullback property” (5.37), and a number of group-theoretic results, which are proved by means of localisation theory. These results are generally not to be found in the cited monograph. The wordings of some of these results do not refer to localisation theory, e.g. 5.31, 5.32, 5.40, 5.41, 5.42, 5.81. As a sample, we cite Theorem 5.81: ”Let G be a finitely generated nilpotent group, let H be a subgroup, write \(H^ N\) for the normal closure of H in G, and let p be a prime. Then the index [G:H] is not divisible by p if and only if this is true for the index \([G:H^ N].''\) Finally, a localisation theory of nilpotent actions of nilpotent groups on nilpotent groups is described at the end of Section 5. In Section 6 the localisation of simply connected CW complexes by means of a cell-by- cell localisation process is presented. In Section 7 the localisation of simply connected CW complexes is generalised to nilpotent ones by means of an inductive construction involving the Postnikov-system. Next, the genus of nilpotent groups and spaces is studied (7.36-7.44), and a somewhat simplified localisation construction for H’- and H-spaces is given (7.47-7.51). The H-space construction was used by Sullivan [1970; loc. cit.] to express the results determining the structure of G/PL. Finally, applications to H-spaces are studied by means of A. Zabrodsky’s technique for ”mixing homotopy types” [Topology 9, 121-128 (1970; Zbl 0191.539)]. As an example, a finite (non-classical) H-space with only 3-torsion is constructed (7.57).
In summary, within its scope, the book presents its subject well, and anyone who wants to enter into the topic and is able to read German will find it useful.