Nilpotente Gruppen und nilpotente Räume. Nachdiplomvorlesung gehalten am Mathematik-Departement ETH Zürich 1981/82. (Aufzeichnungen von Markus Pfenniger.)
type: Item
instance of: scholarly article
title: Nilpotente Gruppen und nilpotente Räume. Nachdiplomvorlesung gehalten am Mathematik-Departement ETH Zürich 1981/82. (Aufzeichnungen von Markus Pfenniger.)
zbMATH Open document ID: 0549.55001
published in: Lecture Notes in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: [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 \textit{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 \textit{M. Lazard} [Ann. Sci. Éc. Norm. Supér., III. Sér. 71, 101-190 (1954; Zbl 0055.251)], and others [\textit{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); \textit{D. Quillen}, Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.537); \textit{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 \textit{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 \textit{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 \textit{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)], \textit{A. K. Bousfield} and \textit{D. M. Kan} [Homotopy limits, completions and loc…
Mathematics Subject Classification ID: 55-01
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 20-01
zbMATH DE Number: 3876198
zbMATH Keywords: localisation of nilpotent groups, localisation of nilpotent homotopy types
zbMATH Keywords: localisation for H-spaces
zbMATH Keywords: co-H-spaces
zbMATH Keywords: Localisation in homotopy theory
zbMATH Keywords: nilpotent actions of nilpotent groups on nilpotent groups
zbMATH Keywords: localisation of simply connected CW complexes
zbMATH Keywords: Postnikov-system
zbMATH Keywords: mixing homotopy types
author: Peter Hilton
reviewed by: J. Huebschmann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0071369
OpenAlex ID: W2500525833
DOI: 10.1007/BFB0071369
description: scientific article; zbMATH DE number 3876198