Small Model $2$-Complexes in $4$-space and Applications

We study finite $2$-complexes built from group presentations, called model $2$-complexes. Model complexes have fundamental group isomorphic with the group presented. We show that there are model complexes of size in the order of bit-complexity of the presentation that can be realized linearly in $\mathbb{R}^4$. We further derive some applications of this result regarding embeddability problems in the euclidean $4$-space, and complexity of computing integral homology groups.