Abstract. We show that if G × M → M is a cohomogeneity one action of a compact connected Lie group G on a compact connected manifold M then H * G (M ) is a Cohen-Macaulay module over H * (BG). Moreover, this module is free if and only if the rank of at least one isotropy group is equal to rank G. We deduce as corollaries several results concerning the usual (de Rham) cohomology of M , such as the following obstruction to the existence of a cohomogeneity one action: if M admits a cohomogeneity one action, then χ(M ) > 0 if and only if H odd (M ) = {0}.

Consider the infinite dimensional flag manifold LK/T corresponding to the simple Lie group K of rank l and with maximal torus T . We show that, for K of type A, B or C, if we endow the space H * (LK/T ) ⊗ R[q 1 , . . . , q l+1 ] (where q 1 , . . . , q l+1 are multiplicative variables) with an R[{q j }]-bilinear product satisfying some simple properties analogous to the quantum product on QH * (K/T ), then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of QH * (K/T ) is determined by the integrals of motion of the non-periodic Toda lattice (see Kim [9]). This is an infinite dimensional extension of the main result of [11] and at the same time a generalization of [6].

Burstall and Guest have given a classification of harmonic maps of the 2‐sphere with values in Lie groups and inner symmetric spaces. We extend their result to outer symmetric spaces G/K, using the pointed Cartan embedding into G. We show that in this case the number of classes can be reduced from 2r to 2s where r = rank G and s = rank K. Moreover we replace the 2‐sphere by a simply connected compact Kähler manifold and ‘harmonic’ by ‘pluriharmonic’.

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We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.