Cohomology of regular spherical actions

Abstract

We study regular spherical actions, namely, actions of a connected compact Lie group G on a manifold M with equidimensional orbits, some of which are spheres. These spherical orbits are diffeomorphic to homogeneous spaces G/H, and there is a list of nine possible families of pairs (G, H). The normalizer N of H in G plays an important role in our calculations, as does the Weyl group W = N/H. In fact, there are three possibilities for W, namely: Z_2, S^1 and S^3. The goal of this paper is to relate the cohomology of M to that of the orbit space M/G. We obtain a particularly simple relationship in the first case, W = Z_2. For the remaining two cases, W = S^1 and W = S^3, it is already known that the cohomology of M and that of the orbit space M/G are related by the Gysin sequence ... \to H^k(M/G) \to H^k(M ) \to H^{k-d}(M/G) \to H^{k+1}(M/G) \to ... In this work, we provide a direct construction of this sequence, avoiding the use of spectral sequences. Moreover, we give an explicit description of the connecting homomorphism as multiplication by a power of the Euler class associated with the action of the Weyl group W = S^l on the fixed-point set M^H.