Abstract
The notion of pure T-structure of rank 2 with singularities is one of the notions introduced by Cheeger and Gromov in [CG1]. It is a generalization of both: a manifold with effective action of a torus T2 and a manifold being the total space of a T2 bundle with Aff(T2) as a structure group. Using well known properties of the group SL2(Z)≅Z6*z2Z4 and a smooth substitute of a classifying map we show that a compact orientable manifold with local T2 action with suitable assumptions on orbit types is equivariantly cobordant with CP2 bundle over a manifold, where T2 acts in a standard way on fibers. The result is an important step towards calculating bordism group of manifolds with mixed singular T-structures. In dimensions 4, 5 and 6 we calculate explicit generators.
