Toric objects associated with the dodecahedron

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

Example of C-rigid polytopes which are not B-rigid

A simple polytope P is said to be B-rigid if its combinatorial structure is characterized by its Tor-algebra, and is said to be C-rigid if its combinatorial structure is characterized by the cohomology ring of a quasitoric manifold over P. It is known that a B-rigid simple polytope is C-rigid. In this paper, we show that the B-rigidity is not equivalent to the C-rigidity.

Topological rigidity of quasitoric manifolds

Quasitoric manifolds are manifolds that admit an action of the torus that is locally the same as the standard action of $T^n$ on $\mathbb{C}^n$. It is known that the quotients of such actions are nice manifolds with corners. We prove that a class of locally standard manifolds, that contains the quasitoric manifolds, is equivariantly rigid, i.e., that any manifold that is $T^n$-homotopy equivalent to a quasitoric manifold is $T^n$-homeomorphic to it.

Immersions and embeddings of quasitoric manifolds over the cube

A quasitoric manifold M2n over the cube In is studied. The Stiefel-Whitney classes are calculated and used as the obstructions for immersions, embeddings and totally skew embeddings. The manifold M2n, when n is a power of 2, has interesting properties: imm(M2n) = 4n ? 2, em(M2n) = 4n ? 1 and N(M2n)? 8n?3.