Equivariant classification of 2-torus manifolds

Abstract

Abstract. We consider locally standard 2-torus manifolds, which are a generalization of small covers of Davis and Januszkiewicz and study their equivariant classification. We formulate a necessary and sufficient condition for two locally standard 2-torus manifolds over the same orbit space to be equivariantly homeomorphic. This leads us to count the equivariant homeomorphism classes of locally standard 2-torus manifolds with the same orbit space.1. Introduction. A 2-torus manifold is a closed smooth manifold of dimension n with a non-free effective action of a 2-torus group (Z 2 ) n of rank n, and it is said to be locally standard if it is locally isomorphic to a faithful representation of (Z 2 ) n on R n . The orbit space Q of a locally standard 2-torus M under this 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 (Z 2 ) n . 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 * ) n where C * = C\{0}. CP n has complex conjugation and its fixed point set is RP n . 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 between 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.The orbit space Q of a locally standard 2-torus manifold M contains a lot of topological information on M . For instance, when Q is a simple convex polytope (in other words, when M is a small cover), the Betti numbers of M (with Z 2 coefficients) are described in terms of face numbers of Q ([4]). This is not the case for a general Q, but the Euler characteristic of M can be described in terms of Q (Lemma 4.1). Although Q contains a lot of

Operations on 3-dimensional small covers

“…In this paper, we are interested in the n = 3 case. First we give some basic facts for 2-torus manifolds (see [5,6] for detail).…”

“…First, we shall recall the terminology: 2-torus manifolds in [5,6]. A 2-torus manifold M n is an n-dimensional, closed smooth manifold with a non-free effective smooth Z n 2 -action.…”

Cohomological rigidity and the number of homeomorphism types for small covers over prisms

“…[9,7,16,[2][3][4]11,12,10,13,14]). In some sense, the classification up to equivariant homeomorphism of small covers over a simple convex polytope has been understood very well.…”

“…In some sense, the classification up to equivariant homeomorphism of small covers over a simple convex polytope has been understood very well. Actually, this can be seen from the following two kinds of viewpoints: One is that two small covers M(λ 1 ) and M(λ 2 ) over a simple convex polytope P n are equivariantly homeomorphic if and only if there is an automorphism h ∈ Aut(P n ) such that λ 1 = λ 2 •h whereh induced by h is an automorphism on all facets of P n (see [11]); the other one is that two small covers M(λ 1 ) and M(λ 2 ) over a simple convex polytope P n are equivariantly homeomorphic if and only if their equivariant cohomologies are isomorphic as H * (B(Z 2 ) n ; Z 2 )-algebras (see [13]). However, in non-equivariant case, the classification up to homeomorphism of small covers over a simple convex polytope is far from understood very well except for few special polytopes (see, e.g.…”

Small covers over products of a simple polytope with a simplex

“…Later, as a generalization of this result, Lü and Masuda investigated a closed smooth manifold of dimension n with a nonfree effective action of (Z 2 ) n [13]. The orbit space Q is a nice manifold with corners.…”