All transitive actions of compact Lie groups on spheres have been determined by D. Montgomery and H. Samelson [Ann. Math. (2) 44, 454-470 (1943; Zbl 0063.04077)] and A. Borel [Bull. Am. Math. Soc. 55, 580-587 (1949; Zbl 0034.01603) and C. R. Acad. Sci., Paris 230, 1378-1380 (1950; Zbl 0041.52203)]. This classification has been extended in various directions, but the case of vanishing Euler characteristic remained inaccessible to general methods.
In the monograph under review, the author determines all \(1\)-connected homogeneous spaces \(G/H\) such that \(G/H\) shares the rational cohomology of a product of spheres \(\mathbb S^m\times\mathbb S^n\), where \(3\leq m\leq n\) and \(n\) is odd. Examples of such spaces (apart from products of spheres) are provided by Stiefel manifolds (of orthonormal \(2\)-frames in real, complex or quaternion spaces), or by certain homogeneous sphere bundles. It is shown (Theorem 3.15) that these three series, together with \(10\) sporadic spaces, cover all examples. As a corollary (3.16), one obtains that only products of spheres and Stiefel manifolds occur for \(m>9\).
For the proof of this result, it is first shown that \(G/H\) even shares the rational homotopy groups with the product of spheres. Explicit information on the rational homotopy of \(G\) and \(H\) allows to determine the Leray-Serre spectral sequence of the principal bundle \(H\to G\to G/H\), and to derive relations between \((m,n)\) and the degrees of the primitive elements in the rational cohomology of \(G\) and \(H\). Moreover, the search for \((G,H)\) is reduced to the search for suitable pairs \((K,K\cap H)\) where \(K\) is a transitive normal subgroup of \(G\), and the Lie algebra of \(K\) is either simple or the direct sum of two simple ideals. The condition \(m\geq 3\) implies that \(K\cap H\) is semisimple, as well. Now representation theory is used to obtain an explicit classification of the pairs \((K,K\cap H)\), and thus the pairs \((G,H)\).
The proof also yields a determination of all transitive actions of compact connected Lie groups on \(1\)-connected rational homology spheres. Thus it contains the known classifications of transitive actions on spheres or on spaces sharing the homology of Stiefel manifolds of real vector spaces of odd dimension.
One of the (motivating) applications of this classification of homogeneous spaces is a question about Tits buildings of rank \(2\), also known as generalized polygons. While spherical buildings of rank greater than \(2\) satisfy the Moufang condition and may be described by BN-pairs in simple algebraic groups, the situation is much more complicated in the rank \(2\) case, and additional assumptions are needed. With an eye on applications to symmetric spaces, Fürstenberg boundaries and isoparametric submanifolds, one stipulates compact connected topologies, and a transitive group action. The compact topologies are well understood, the author has determined the cohomology in his dissertation [Compact polygons. Tübingen: Math. Fak., Univ. Tübingen, vi, 72 p. (1994; Zbl 0844.51006)]. In particular, there are various restrictions on the topological parameters \((\dim D_1(\ell),\dim D_1(p))\), where \(p\) is a point and \(\ell\) is a line.
N. Knarr [Forum Math. 2, 603-612 (1990; Zbl 0711.51002)] has proved an analogon to the Feit-Higman theorem [W. Feit and G. Higman, J. Algebra 1, 114-131 (1964; Zbl 0126.05303)] restricting the possible values of \(n\) for compact connected generalized \(n\)-gons to \(n\in\{3,4,6\}\). Previous results by H. Salzmann [Pac. J. Math. 60, 217-234 (1975; Zbl 0323.50009)] and R. Löwen [J. Reine Angew. Math. 321, 217-220 (1981; Zbl 0443.51012)] have treated the case of point-homogeneous compact connected generalized triangles (projective planes). The case of hexagons has been covered in the author’s dissertation [loc. cit.], and there remains the case of generalized quadrangles. The results culminate in Theorem 7.33: Consider a point-homogeneous compact connected generalized quadrangle \(\mathcal Q\) with topological parameters \((m,n):=\left(\dim D_1(\ell),\dim D_1(p)\right)\) and \(m\geq 2\). Then there is a point-transitive compact connected subgroup \(G\) of the group of all automorphisms of \(\mathcal Q\). If \((m,n)\) satisfies one of the following conditions, the quadrangle is uniquely determined: \(m=n\) (then \(\mathcal Q\) is the symplectic quadrangle over the reals or the complex numbers, up to duality), \((m,n)=(4,4s-5)\) for \(s\geq 4\) (then \(\mathcal Q\) is a hermitian quadrangle over the quaternions), \((m,n)=(4s-5,4)\) for \(s\geq 4\) (dual to the previous case), \((m,n)=(2s-3,2)\) for \(s\geq 5\) (duals of complex hermitian quadrangles), \((m,n)=(s-1,1)\) for \(s\geq 9\) (duals of real orthogonal quadrangles). In the remaining cases, the author determines at least the homogeneous space \(G/H\) (where \(H\) is a point stabilizer in \(G\)), using Theorem 3.15. An interesting corollary states that \(m>9\) implies that \(\mathcal Q\) is a Moufang quadrangle, and is known explicitly.
In some of the cases, one considers topological parameters that do not occur with any known generalized quadrangle. Indeed, deep results by S. Stolz [Invent. Math. 138, 253-279 (1999; Zbl 0944.53035)] have been used by R. Markert [Isoparametric hypersurfaces and generalized quadrangles, Diplomarbeit, Math. Institut Univ. Würzburg (1999)] to restrict the possible parameters. However, the present treatment does not depend on these additional restrictions.
A further application of the classification of quadrangles (Theorem 7.33, via the classification of transitive actions, Theorem 3.15) belongs to the geometry of submanifolds (Theorem 8.15): Let \(M\) be an isoparametric hypersurface in a sphere such that the isometry group acts transitively on at least one of the focal manifolds. Then either \(M\) itself is homogeneous (and known explicitly), or it is of Clifford type. In particular, the author obtains a new, independent proof of the classification of homogeneous isoparametric hypersurfaces given by W.-J. Hsiang and H. B. Lawson jun. [J. Differ. Geom. 5, 1-38 (1971; Zbl 0219.53045)].
Contents: 1. The Leray-Serre spectral sequence. 2. Ranks of homotopy groups. 3. Some homogeneous spaces. 4. Representations of compact Lie groups. 5. The case when \(G\) is simple. 6. The case when \(G\) is semisimple. 7. Homogeneous compact quadrangles. 8. Homogeneous focal manifolds.