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Associated with the Davis-Januszkiewicz theory of small covers, this paperdeals with the theory of 2-torus manifolds from the viewpoint of equivariantbordism. We define a differential operator on the "dual" algebra of theunoriented $G_n$-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), wemay use this differential operator to give a very simple description of tomDieck-Kosniowski-Stong localization theorem in the setting of 2-torusmanifolds. We then apply this to study the $G_n$-equivariant unoriented bordismclassification of $n$-dimensional 2-torus manifolds. We show that the$G_n$-equivariant unoriented bordism class of each $n$-dimensional 2-torusmanifold contains an $n$-dimensional small cover as its representative, solvingthe conjecture posed in [19]. In addition, we also obtain that the gradednoncommutative ring formed by the equivariant unoriented bordism classes of2-torus manifolds of all possible dimensions is generated by the classes of allgeneralized real Bott manifolds (as special small covers over the products ofsimplices). This gives a strong connection between the computation of$G_n$-equivariant bordism groups or ring and the Davis-Januszkiewicz theory ofsmall covers. As a computational application, with the help of computer, wecompletely determine the structure of the group formed by equivariant bordismclasses of all 4-dimensional 2-torus manifolds. Finally, we give some essentialrelationships among 2-torus manifolds, coloring polynomials, colored simpleconvex polytopes, colored graphs.

Small covers and the equivariant bordism classification of 2-torus manifolds