##
**Compact projective planes. With an introduction to octonion geometry.**
*(English)*
Zbl 0851.51003

De Gruyter Expositions in Mathematics. 21. Berlin: de Gruyter. xiii, 688 p. (1996).

A topological projective plane is a projective plane \({\mathcal P} = (P,{\mathcal L})\) in which the point set \(P\) and the set of lines \(\mathcal L\) are equipped with topologies that are neither discrete nor indiscrete such that the geometric operations of joining two points by a line and of intersecting two lines in a point are continuous on their domains of definition. One calls such a topological projective plane (locally) compact, connected or \(n\)-dimensional if the point space has the respective property. The study of topological projective planes involves tools and techniques from many different areas of mathematics and the book gives an excellent illustration of their interplay.

The study of compact, connected topological projective planes was initiated in the early 1950s by H. Salzmann. Since then many beautiful and deep results have been achieved by an ever growing number of researchers in this area. In particular, 2- and 4-dimensional planes are now well understood and the theory of 8- and 16-dimensional planes is rapidly developing. However, the results are scattered in the literature. In that this book gives a systematic account of most of these results it has been long waited for.

The book provides a wealth of information and should prove an extremely useful text both as an introduction to compact, connected topological projective planes as well as a valuable and convenient reference and a sound foundation for future investigations. It gives an up-to-date picture incorporating unpublished results but also including simplified proofs and proofs of folklore.

The book is a highly readable, self-contained monograph. It serves as an excellent advertisement for topological geometry and will attract many more mathematicians to this area. Various chapters may appeal to the interested broader mathematical community. In particular, those who are interested in topology and the theory of Lie groups and their interactions in other disciplines may consult the text under review. The book is divided into nine chapters and includes an extensive bibliography covering the research in compact topological projective planes since its inception.

Chapter 1 gives detailed descriptions of the classical planes, i.e., the desarguesian planes over the real numbers \(\mathbb{R}\), complex numbers \(\mathbb{C}\) and Hamilton’s quaternions \(\mathbb{H}\) and the Moufang plane over Cayley’s octonions \(\mathbb{O}\). Both geometric and topological properties of these planes are investigated. Particular care is given to the construction of the projective plane \({\mathcal P}_2(\mathbb{O})\) over \(\mathbb{O}\) since \(\mathbb{O}\) is not a division ring. This section also includes the determination and a detailed study of the collineation group of \({\mathcal P}_2 (\mathbb{Q})\) which is the real exceptional simple Lie group \(E_6(-26)\). All this is done from an incidence geometric point of view with only marginal references to Lie theory. Furthermore, the exceptional simple Lie groups \(F_4(-20)\) and \(F_4(-52)\) can be found in the collineation group. In this way an easy access to some of the more elusive exceptional simple Lie groups is provided which may be of interest to those working in Lie theory.

The following chapter gives a brief summary of the basic background on projective planes, the coordinatization process in and collineations of such planes. In particular, the fundamental structure of translation planes and their associated ternary fields is reviewed.

In Chapter 3 the origins of topological incidence geometry are revisited. Geometries with point set \(\mathbb{R}^3\) and lines closed subsets thereof homeomorphic to \(\mathbb{R}\) and geometries on surfaces (i.e., topological manifolds of dimension 2) are studied. Various characterizations of the real desarguesian plane are given. Furthermore, all 2-dimensional projective planes whose collineation groups have dimensions at least 3 are classified.

Chapter 4 provides the foundation for the study of compact, connected, finite-dimensional projective planes through their collineation groups. In the first section of this chapter topological properties of lines, the point sets and the line sets of topological locally compact projective planes are derived. Later, connectedness of the point set is added. All compact, connected Moufang planes are classified. (These are the classical planes introduced in Chapter 1.) Then the set of all continuous collineations \(\Sigma\) of such a plane is studied. \(\Sigma\) is a group and when endowed with the compact-open topology \(\Sigma\) becomes a locally compact group with countable basis, acting as a topological transformation group on the point space, line space and flag space. Later in the book it it shown that \(\Sigma\) is even a Lie group if the topological dimension of the point set is 2 or 4 or if its topological dimension is large enough or if various other conditions are satisfied.

Chapter 5 deals with compact, connected, finite-dimensional projective planes from the point of view of algebraic topology. Many of the invariants of algebraic topology are computed for these planes. One always obtains the same invariants as for the classical counterparts. In particular, the point space of such a plane has dimension \(2l\) where \(l\) is one of the numbers 1, 2, 4 or 8. \(l\) is the dimension of a line in such a plane and each line is homotopy equivalent to an \(l\)-sphere.

In the following chapter various homogeneity conitions in compact, connected projective planes are investigated. In particular, groups of axial collineations and transitivity conditions for these groups are studied. Furthermore, the classical planes are characterized with respect to these groups and as compact projective planes admitting point-transitive collineation groups. This chapter also reviews the known results about groups of projectivities of compact, connected projective planes.

Chapters 7 and 8 give the state of knowledge on the classification of compact projective planes according to the dimensions of their collineation groups. Chapter 7 deals with the 4-dimensional planes and the next chapter with the 8- and 16-dimensional cases. 4-dimensional projective planes stand out among the higher-dimensional projective planes by the existence of shift planes, which are investigated in detail in one section of this chapter and which provide plenty of examples of nontranslation planes. All shift planes admitting collineation groups of dimension at least 6 are known. Furthermore, it is shown that 4-dimensional projective planes with a collineation group of dimension at least 7 are translation planes or dual translation planes or the single shift plane of Knarr. Various characterizations of the classical complex projective planes among the 4-dimensional projective planes are given.

Translation planes again have a prominent role among 8- and 16-dimensional projective plane. The translation group of such a plane is an \(\mathbb{R}^8\) or \(\mathbb{R}^{16}\) respectively. The stabilizer of a point not on the translation line is linearly represented on the corresponding \(\mathbb{R}^n\) which links the investigation of these planes to the theory of Lie groups and their representations. However, unlike as for 4-dimensional translation planes, these stabilizers are often neither solvable nor semisimple. This fact and the difficulty of mastering the action on the translation line account for many of the problems in the investigation of 8- and 16-dimensional translation planes. The prevalence of translation planes further shows in that they or their duals appear among the most homogeneous planes and that most known examples of 8- and 16-dimensional projective planes are of this kind.

A detailed study of 8- and 16-dimensional Hughes planes is also included in this chapter. (These are examples of planes that are neither translation planes nor dual translation planes.) Furthermore all such planes are classified and they are characterized as the only compact, non-Moufang planes of dimension \(4d\) admitting an action of a group locally isomorphic to the linear collineation group of the desarguesian plane of dimension \(2d\).

In the last chapter useful tools from topology and Lie theory are collected. This includes a characterization of Lie groups as certain topological groups and the fundamental structure and classification of almost simple Lie groups and their real linear representations. In particular, almost simple Lie groups of dimensions \(\leq 52\) and their irreducible linear representations on real vector spaces of dimensions \(\leq 16\) are listed.

The study of compact, connected topological projective planes was initiated in the early 1950s by H. Salzmann. Since then many beautiful and deep results have been achieved by an ever growing number of researchers in this area. In particular, 2- and 4-dimensional planes are now well understood and the theory of 8- and 16-dimensional planes is rapidly developing. However, the results are scattered in the literature. In that this book gives a systematic account of most of these results it has been long waited for.

The book provides a wealth of information and should prove an extremely useful text both as an introduction to compact, connected topological projective planes as well as a valuable and convenient reference and a sound foundation for future investigations. It gives an up-to-date picture incorporating unpublished results but also including simplified proofs and proofs of folklore.

The book is a highly readable, self-contained monograph. It serves as an excellent advertisement for topological geometry and will attract many more mathematicians to this area. Various chapters may appeal to the interested broader mathematical community. In particular, those who are interested in topology and the theory of Lie groups and their interactions in other disciplines may consult the text under review. The book is divided into nine chapters and includes an extensive bibliography covering the research in compact topological projective planes since its inception.

Chapter 1 gives detailed descriptions of the classical planes, i.e., the desarguesian planes over the real numbers \(\mathbb{R}\), complex numbers \(\mathbb{C}\) and Hamilton’s quaternions \(\mathbb{H}\) and the Moufang plane over Cayley’s octonions \(\mathbb{O}\). Both geometric and topological properties of these planes are investigated. Particular care is given to the construction of the projective plane \({\mathcal P}_2(\mathbb{O})\) over \(\mathbb{O}\) since \(\mathbb{O}\) is not a division ring. This section also includes the determination and a detailed study of the collineation group of \({\mathcal P}_2 (\mathbb{Q})\) which is the real exceptional simple Lie group \(E_6(-26)\). All this is done from an incidence geometric point of view with only marginal references to Lie theory. Furthermore, the exceptional simple Lie groups \(F_4(-20)\) and \(F_4(-52)\) can be found in the collineation group. In this way an easy access to some of the more elusive exceptional simple Lie groups is provided which may be of interest to those working in Lie theory.

The following chapter gives a brief summary of the basic background on projective planes, the coordinatization process in and collineations of such planes. In particular, the fundamental structure of translation planes and their associated ternary fields is reviewed.

In Chapter 3 the origins of topological incidence geometry are revisited. Geometries with point set \(\mathbb{R}^3\) and lines closed subsets thereof homeomorphic to \(\mathbb{R}\) and geometries on surfaces (i.e., topological manifolds of dimension 2) are studied. Various characterizations of the real desarguesian plane are given. Furthermore, all 2-dimensional projective planes whose collineation groups have dimensions at least 3 are classified.

Chapter 4 provides the foundation for the study of compact, connected, finite-dimensional projective planes through their collineation groups. In the first section of this chapter topological properties of lines, the point sets and the line sets of topological locally compact projective planes are derived. Later, connectedness of the point set is added. All compact, connected Moufang planes are classified. (These are the classical planes introduced in Chapter 1.) Then the set of all continuous collineations \(\Sigma\) of such a plane is studied. \(\Sigma\) is a group and when endowed with the compact-open topology \(\Sigma\) becomes a locally compact group with countable basis, acting as a topological transformation group on the point space, line space and flag space. Later in the book it it shown that \(\Sigma\) is even a Lie group if the topological dimension of the point set is 2 or 4 or if its topological dimension is large enough or if various other conditions are satisfied.

Chapter 5 deals with compact, connected, finite-dimensional projective planes from the point of view of algebraic topology. Many of the invariants of algebraic topology are computed for these planes. One always obtains the same invariants as for the classical counterparts. In particular, the point space of such a plane has dimension \(2l\) where \(l\) is one of the numbers 1, 2, 4 or 8. \(l\) is the dimension of a line in such a plane and each line is homotopy equivalent to an \(l\)-sphere.

In the following chapter various homogeneity conitions in compact, connected projective planes are investigated. In particular, groups of axial collineations and transitivity conditions for these groups are studied. Furthermore, the classical planes are characterized with respect to these groups and as compact projective planes admitting point-transitive collineation groups. This chapter also reviews the known results about groups of projectivities of compact, connected projective planes.

Chapters 7 and 8 give the state of knowledge on the classification of compact projective planes according to the dimensions of their collineation groups. Chapter 7 deals with the 4-dimensional planes and the next chapter with the 8- and 16-dimensional cases. 4-dimensional projective planes stand out among the higher-dimensional projective planes by the existence of shift planes, which are investigated in detail in one section of this chapter and which provide plenty of examples of nontranslation planes. All shift planes admitting collineation groups of dimension at least 6 are known. Furthermore, it is shown that 4-dimensional projective planes with a collineation group of dimension at least 7 are translation planes or dual translation planes or the single shift plane of Knarr. Various characterizations of the classical complex projective planes among the 4-dimensional projective planes are given.

Translation planes again have a prominent role among 8- and 16-dimensional projective plane. The translation group of such a plane is an \(\mathbb{R}^8\) or \(\mathbb{R}^{16}\) respectively. The stabilizer of a point not on the translation line is linearly represented on the corresponding \(\mathbb{R}^n\) which links the investigation of these planes to the theory of Lie groups and their representations. However, unlike as for 4-dimensional translation planes, these stabilizers are often neither solvable nor semisimple. This fact and the difficulty of mastering the action on the translation line account for many of the problems in the investigation of 8- and 16-dimensional translation planes. The prevalence of translation planes further shows in that they or their duals appear among the most homogeneous planes and that most known examples of 8- and 16-dimensional projective planes are of this kind.

A detailed study of 8- and 16-dimensional Hughes planes is also included in this chapter. (These are examples of planes that are neither translation planes nor dual translation planes.) Furthermore all such planes are classified and they are characterized as the only compact, non-Moufang planes of dimension \(4d\) admitting an action of a group locally isomorphic to the linear collineation group of the desarguesian plane of dimension \(2d\).

In the last chapter useful tools from topology and Lie theory are collected. This includes a characterization of Lie groups as certain topological groups and the fundamental structure and classification of almost simple Lie groups and their real linear representations. In particular, almost simple Lie groups of dimensions \(\leq 52\) and their irreducible linear representations on real vector spaces of dimensions \(\leq 16\) are listed.

Reviewer: Günter F. Steinke (Christchurch)

### MSC:

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |

51H10 | Topological linear incidence structures |

51H25 | Geometries with differentiable structure |

51A40 | Translation planes and spreads in linear incidence geometry |

51N30 | Geometry of classical groups |