Equivariant cohomology ring of open torus manifolds with locally standard actions

Abstract: The notation of torus manifolds were introduced by A. Hattori and M. Masuda. Toric manifolds, quasitoric manifolds, topological toric manifolds, toric origami manifolds and $b$-symplectic toric manifolds are typical examples of torus manifolds with locally standard action. Recently, L. Yu introduced a nice notion topological face ring $\mathbf{k}[Q]$, a generalization of Stanley-Reisener ring, for a nice manifold with corners $Q$. L. Yu applied polyhedral product technique developed by A. Bahri, M. Bendersky, F. Cohon and S. Gilter to show that the equivariant cohomology ring $H^*_T(M)$ of an open torus manifold $M$ with locally standard action is isomorphic to the topological face ring of $M/T$ under the assumption that the free part of the action is a trivial torus bundle. In this paper we show that Yu's formula holds for any open torus manifolds with locally standard action by a different appoach. In addition using our method we give an explicit formula for equivariant Stiefel-Whitney classes and Pontrjagin classes of open torus manifolds with locally standard action.