New Constructions related to the Polynomial Sphere Recognition Problem

Abstract

We construct a simply connected $2-$complex $C$ embeddable in $3-$space such that for any embedding of $C$ in ${\mathbb{S}}^3$, any edge contraction forms a minor of the $2-$complex not embeddable in $3-$space. We achieve this by proving that every edge of $C$ forms a nontrivial knot in any of the embeddings of $C$ in ${\mathbb{S}}^3$.

22 pages, 11 figures