Some results on the topology of quasitoric manifolds and their equivariant mapping spaces

We apply methods of homotopy theory to the study of quasitoric manifolds. More specifically, we determine a simple criterion for rational ellipticity of a quasitoric manifold based on the combinatorics of the orbit polytope. We also study the topology of some equivariant mapping spaces of the quasitoric manifolds and their associated moment angle complexes. In case the image polytope is a product of simplices we completely determine the homotopy type of the mapping spaces in question. We also suggest a way to study the topology of the equivariant mapping spaces for a general simple polytope using the Bousfield- Kan spectral sequence. As an application we derive some connectivity results for equivariant mapping spaces of manifolds over 2-dimensional polytope