Hirzebruch genera of quasitoric manifolds

Abstract

A quasitoric manifold $M^{2n}$ is a smooth compact manifold with a locally standard $T^n$ -action whose orbit space is diffeomorphic to a combinatorial simple polytope as manifolds with corners. Then the relative interior points in a k-face of $P^n$ correspond to the orbits with the same isotropy subgroup of codimension k. We give a stably complex structure on a quasitoric manifold from a given omniorientation of the manifold. From the relation between quasitoric manifolds and the corresponding polytope, we obtain the formula for Hirzebruch genera of quasitoric manifolds only using the combinatorial data. We then calculate the Hirzebruch genera of quasitoric manifolds over a triangle and a square.