Note on Totally Skew Embeddings of Quasitoric Manifolds over Cube

Totally skew embeddings are introduced by Ghomi and Tabachnikov. They are naturally related to classical problems in topology, such as the generalized vector field problem and the immersion problem for real projective spaces. In recent paper Topological obstructions to totally skew embeddings, {totaly skew} embeddings are studied by using the Stiefel-Whitney classes In the same paper it is conjectured that for every $n$-dimensional, compact smooth manifold $M^n$ $(n>1)$, $$N(M^n)\leq 4n-2\alpha (n)+1,$$ where $N(M^n)$ is defined as the smallest dimension $N$ such that there exists a {\em totally skew} embedding of a smooth manifold $M^n$ in $\mathbb{R}^N$. We prove that for every $n$, there is a quasitoric manifold $Q^{2n}$ for which the orbit space of $T^n$ action is a cube $I^n$ and $$N(Q^{2n})\geq 8n-4\alpha (n)+1.$$ Using the combinatorial properties of cohomology ring $H^* (Q^{2n}, \mathbb{Z}_2)$, we construct an interesting general non-trivial example different from known example of the product of complex projective spaces.