Note on Totally Skew Embeddings of Quasitoric Manifolds over Cube

TL;DR

This paper investigates totally skew embeddings of quasitoric manifolds over cubes, providing new lower bounds and constructing explicit examples that differ from classical cases, thus advancing understanding in topological embedding problems.

Contribution

It establishes a new lower bound for the dimension of totally skew embeddings of certain quasitoric manifolds and constructs explicit examples distinct from known cases.

Findings

Proves lower bound N(Q^{2n}) ≥ 8n - 4α(n) + 1 for specific quasitoric manifolds.

Constructs a non-trivial example of a quasitoric manifold over a cube with unique cohomological properties.

Links topological obstructions to combinatorial properties of cohomology rings.

Abstract

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.