Ring Surface Topology

This thesis focuses on the 2-torus(Z₂^k) topology.The first work is motivated by the GKM theory([32]).We focus on a class of connected closed Z₂-manifolds M,which have nonempty fixed points sets,and the number of the fixed points is finite,equal to the sum of the mod 2 Betti numbers of M.Applying the famous theorem—localization theorem,and the technique of equivariant index,we proved the ring structure theorem of the equivariant cohomology of M.After proving the ring isomorphism theorem,we discuss the number of equivalent classes of equivariant cohomology rings in the cases of dimension 2 and 3.For the dimension 2 case,we obtained a complete answers.While the case of dimension 3 is much more complicated.We gave a method of generating a class of the equivariant cohomology,which are 1 to 1 correspond to the binary linear self-dual codes in the coding theory,as informed by professor Puppe.The second work is a generalization of the work of Davis and Januszkiewicz( [21]),where the manifolds admit locally standard actions,and the orbit spaces are convex simple polytopes.We only focus on the locally standard action.Note that the orbit spaces of such actions are of corner structures as the simple polytopes have, as well as a coloring map on the faces of the corner structures.More interesting thing is that we could construct the manifolds totally from the orbit space—the manifolds with corners,the colorings and the principle bundles over the orbit spaces.Applying the technique as shown in[58]and[60],we could reduce any solid 3-ball to a special orbit space,whose associated manifold is the 3-sphere with locally standard action generated by axes reflections.For the case of solid torus S¹Ã—B²,the situation is much more complicated. After introducing two more operations,we could reduce any colored nice solid torus with corners to four elementary ones.We also computed the homology groups of the maifolds constructed by the elementary solid torus and the trivial principle bundles.