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.