Characterization of homogeneous torus manifolds (Q964023)

A torus manifold is an oriented manifold \(M^{2n}\) acted on by a torus \(T^n\) such that both the intersection of all isotropy groups \(\cap_{x\in M} T^n_x\) and the fixed point set \(M^T\) are finite. This paper is the first one in a series of papers that investigates when the torus action can be extended to a homogeneous (transitive) \(G\)-action for a compact, connected Lie group \(G\) whose maximal torus is \(T^n\). In particular the author characterises codimension 0 extended \(G\)-actions up to essential isomorphisms. As \(M\) and \(G\) are compact, if \((M^{2n},G)\) is a codimension 0 extended \(G\)-action then the \(G\)-action on \(M^{2n}\) is transitive and \(M^{2n}\) is a homogeneous manifold. The main classification theorem of this paper states that a homogeneous torus manifold \(M^{2n}\) is (weak equivariantly) diffeomorphic to a product of complex projective spaces \(\prod \mathbb CP(l)\) and quotient spaces of a product of spheres \((\prod S^{2m})/\mathcal A\) with standard torus actions, where \(\mathcal A\) is a subgroup of \(\prod \mathbb Z_2\) generated by the antipodal involutions on \(S^{2m}\).