xiv INTRODUCTION
that the group action has to be linear. Therefore representation theory has to
be replaced by arguments about compact transformation groups acting on locally
compact spaces. The result is as follows.
Theorem Let 0 be a compact connected quadrangle. Assume that the point space
is 9-connected, and that the automorphism group is point transitive. Then 0 is
a Moufang quadrangle (in fact the dual of a classical quadrangle associated to a
hermitian form over R, C or MJ.
At this point, we should mention the following related results by Grundhofer, Knarr
and the author.
Theorem Let A be a compact connected irreducible spherical building of rank at
least 2. Assume that the automorphism group is chamber transitive. Then A is the
Moufang building associated to a simple non-compact Lie group G. If H C Aut(A)
is a connected chamber transitive subgroup, then either H = G, or H is compact.
All compact connected chamber transitive groups in the theorem above were deter-
mined by Eschenburg and Heintze. Combining these results, we have the following
theorem.
Theorem Let A be a compact connected irreducible spherical building of rank k 2.
Assume that the automorphism group is transitive on one type of vertices of A. If
the building is of type Ci assume in addition that either (1) the vertex space in
question is 9-connected, or (2) that the two vertex sets have the same dimension, or
(3) that the action is chamber transitive. Then A is a Moufang building associated
to a simple non-compact real Lie group of real rank k.
In the C2-case, the assumption on the connectivity cannot be dropped completely,
since there are counterexamples which are not highly connected. In view of the
results in the present book, the following conjecture for the C^-case is very natural.
Conjecture Let (5 be a compact connected generalized quadrangle. Assume that
the automorphism group is point or line transitive. The either 0 is a Moufang
quadrangle, or 0 is of Clifford type with topological parameters (3,4/c) or (7,8).
Finally, I should mention here the following new and beautiful result by Immervoll:
every isoparametric hypersurface with g 4 distinct principal curvatures is a C2-
building.
* * *
The material is organized as follows. In the first chapter we collect some well-
known facts about the algebraic topology of fibrations, spectral sequences, and
Eilenberg-MacLane spaces. This chapter should be accessible for any reader with
basic knowledge about algebraic topology. In Chapter 2 we prove an extension of
the Cartan-Serre theorem about rational homotopy groups by a standard homotopy
theoretic method. This is applied in Chapter 3 in order to determine the rational
Leray-Serre spectral sequence associated to the transitive group action.
All facts about representations of compact Lie groups on real, complex or
quaternionic vector spaces which are used in the classification are presented in
Chapter 4, which also contains tables about compact almost simple Lie groups and
their low-dimensional representations. These facts may be of some independent
interest.
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