Mathematics – Algebraic Topology
Scientific paper
2010-08-12
Mathematics
Algebraic Topology
20 pages, updated version with subsection 5.2 modified particularly
Scientific paper
We define a differential operator on the ``dual'' algebra of the unoriented $G$-representation algebra introduced by Conner and Floyd, where $G=(\Z_2)^n$. With the help of $G$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simply equivalent description of tom Dieck--Kosniowski--Stong localization theorem in the setting of smooth closed $n$-manifolds with effective smooth $G$-actions (also called $n$-dimensional 2-torus manifolds). We then apply this to study the $G$-equivariant unoriented cobordism classification of $n$-dimensional 2-torus manifolds. We show that the $G$-equivariant unoriented cobordism class of each $n$-dimensional 2-torus manifold contains an $n$-dimensional small cover as its representative, solving the conjecture posed in [15]. In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented cobordism classes of all possible dimensional 2-torus manifolds is generated by the classes of small covers over the products of simplices.
Lü Zhi
Tan Qiangbo
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