Topological geometry. 2nd ed.

*(English)*Zbl 0446.15001
Cambridge etc.: Cambridge University Press. viii, 486 p. hbk: £25.00; pbk: £9.95 (1981).

The first edition of this book was published by Van Nostrand Reinhold Company in 1969. For the second edition, a few misprints and minor errors have been corrected. The main text also contains a number of additions and improvements. It is a remarkable fact that these modifications have been performed without changing the paging; thus, comparing the two editions is an easy task. The main addition is a new chapter 21.

A sufficiently detailed review for the first edition was written by J. Dieudonné [Zbl 0186.06304]; here we shall only add a few words about chapter 21. This chapter is based on the discussion in chapters 13 and 14 of the Clifford algebras and the Cayley algebra and on that in chapter 20 of the Lie groups and Lie algebras. It is devoted to triality, a feature which occurs in the group \(\mathrm{Spin}\, 8\) and which is the ultimate cause for the appearance of a variety of exceptional phenomena; in fact triality is an automorphism of order three of \(\mathrm{Spin}\, 8\) that does not project to an automorphism of \(\mathrm{SO}(8)\). The details in this chapter include some important transitive actions on spheres, a description of the exceptional fourteen-dimensional Lie group \(G_2\) (the group of automorphisms of the Cayley algebra), and some applications to six-dimensional projective quadrics.

The bibliography also has been updated and completed.

The book continues to be of great interest both for teachers and researchers. Many of its chapters still have no counterparts in other textbooks. Both the author and the publishing house should be praised for the effort to make this text accessible to a wider audience.

A sufficiently detailed review for the first edition was written by J. Dieudonné [Zbl 0186.06304]; here we shall only add a few words about chapter 21. This chapter is based on the discussion in chapters 13 and 14 of the Clifford algebras and the Cayley algebra and on that in chapter 20 of the Lie groups and Lie algebras. It is devoted to triality, a feature which occurs in the group \(\mathrm{Spin}\, 8\) and which is the ultimate cause for the appearance of a variety of exceptional phenomena; in fact triality is an automorphism of order three of \(\mathrm{Spin}\, 8\) that does not project to an automorphism of \(\mathrm{SO}(8)\). The details in this chapter include some important transitive actions on spheres, a description of the exceptional fourteen-dimensional Lie group \(G_2\) (the group of automorphisms of the Cayley algebra), and some applications to six-dimensional projective quadrics.

The bibliography also has been updated and completed.

The book continues to be of great interest both for teachers and researchers. Many of its chapters still have no counterparts in other textbooks. Both the author and the publishing house should be praised for the effort to make this text accessible to a wider audience.

Reviewer: Jack Weinstein

##### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |