Contents
Introduction x
Chapter 1. The Leray-Serre spectral sequence 1
l.A. Additive structure 2
l.B. Multiplicative structure 6
l.C. Notes on collapsing 9
Chapter 2. Ranks of homotopy groups 11
2.A. The Whitehead tower 11
2.B. The Cartan-Serre Theorem 15
Chapter 3. Some homogeneous spaces 18
3.A. Structure of compact Lie groups 18
3.B. Certain homogeneous spaces 21
3.C. The integral classification 28
Chapter 4. Representations of compact Lie groups 33
4.A. The classification of irreducible representations 33
4.B. Subgroups of classical groups 35
4.C. Useful formulas 36
4.D. The Dynkin index 49
Chapter 5. The case when G is simple 51
5.A. Case (I): H # (X) = /\z(u,v). 51
5.B. Case (II): H # (X) = Z[a]/(a2) 8 Az(^). 5 7
Chapter 6. The case when G is semisimple 63
6.A. The split case 63
6.B. The non-split case (I): H*(X) = f\z{u,v). 67
6.C. The non-split case (II): H
#
(X) =
Z[a]/(a2)
g A z M -
7 2
Chapter 7. Homogeneous compact quadrangles 75
7.A. Generalized quadrangles and group actions 77
7.B. Compact quadrangles 79
7.C. Some results about compact transformation groups 83
7.D. Group actions on compact quadrangles 85
7.E. The Stiefel manifolds 87
7.F. The (4, An - 5)-series 90
7.G. Products of spheres 91
7.H. Summary 92
Chapter 8. Homogeneous focal manifolds 94
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