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Topological Classification of Multiaxial U(n)-Actions

A U(n)-manifold is multiaxial if the isotropy groups are always conjugate to unitary subgroups. The classification and the concordance of such manifolds have been studied by Davis, Hsiang and Morgan under much more strict conditions. We show that in general, without much extra condition, the homotopy classification of multiaxial manifolds can be split into a direct sum of the classification of pairs of adjacent strata, which can be computed by the classical surgery theory. Moreover, we also compute the homotopy classification for the case of the standard representation sphere. We also present the result for the similar multiaxial Sp(n)-manifolds.