Small Covers and the Equivariant Bordism Classification of 2-torus Manifolds

Abstract

Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented bordism classification of $n$-dimensional 2-torus manifolds. We show that the $G_n$-equivariant unoriented bordism class of each $n$-dimensional 2-torus manifold contains an $n$-dimensional small cover as its representative, solving the conjecture posed in [19]. In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented bordism classes of 2-torus manifolds of all possible dimensions is generated by the classes of all generalized real Bott manifolds (as special small covers over the products of simplices). This gives a strong connection between the computation of $G_n$-equivariant bordism groups or ring and the Davis-Januszkiewicz theory of small covers. As a computational application, with the help of computer, we completely determine the structure of the group formed by equivariant bordism classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential relationships among 2-torus manifolds, coloring polynomials, colored simple convex polytopes, colored graphs.