Equivariant Classification of 2-torus Manifolds

In this paper, we consider the equivariant classification of locally standard 2-torus manifolds. A 2-torus manifold is a closed smooth manifold of dimension n with an effective action of a 2-torus group (Z2) n of rank n, and it is said to be locally standard if it is locally isomorphic to a faithful representation of (Z2) n on R. The orbit space Q of a locally standard 2-torus M by the action is a nice manifold with corners. When Q is a simple convex polytope, M is called a small cover and studied in [4]. A typical example of a small cover is a real projective space RP n with a standard action of (Z2) . Its orbit space is an n-simplex. On the other hand, a typical example of a compact non-singular toric variety is a complex projective space CP n with a standard action of (C) where C = C\{0}. CP n has complex conjugation and its fixed point set is RP . More generally, any compact non-singular toric variety admits complex conjugation and its fixed point set often provides an example of a small cover. Similarly to the theory of toric varieties, an interesting connection among topology, geometry and combinatorics is discussed for small covers in [4], [5] and [7]. Although locally standard 2-torus manifolds form a much wider class than small covers, one can still expect such a connection. See [9] for the study of 2-torus manifolds from the viewpoint of cobordism.