A quasitoric manifold is a smooth manifold with a good action of a compact torus for which the orbit space is a convex polytope. The most fundamental property of the quasitoric manifolds is that they are in one-to-one correspondence with the characteristic pairs, a kind of combinatorial objects, and we can get the classification up to (weakly) equivariant homeomorphism through this correspondence. However, when one try to advance to the classification up to homeomorphism, it turns out to be difficult since the combinatirial data seem to tell us almost nothing on the homeomorphisms which do not respect the torus actions. On the other hand, an interesting method was introduced in the joint work with Kuwata, Masuda, and Park on the classification of toric manifolds over a cube with one vertex cut, which gives a new way to construct homeomorphisms which are not (weakly) equivariant.

In this talk, we try to extend this method to more general situations from a homotopical viewpoint.