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

TL;DR

This paper demonstrates that computing the first homology group of embeddable 2-complexes in 4-space is as computationally hard as matrix diagonalization, with implications for embeddability problems.

Contribution

It establishes the equivalence of homology computation to matrix diagonalization for 2-complexes in 4-space, extending previous work and constructing model complexes from group presentations.

Findings

Homology computation is as hard as matrix diagonalization.

Model complexes can be realized linearly in 4-space.

Results impact embeddability decision problems in 4-space.

Abstract

We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) $2$-complexes. We show, as an extension of an earlier work, that computing first homology of $2$-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usual procedures for computing homology cannot be improved other than by matrix methods. This is true even if the complex is in the euclidean $4$-space. For this purpose, we use $2$-complexes built in a standard way from group presentations, called model $2$-complexes. Model complexes have fundamental group isomorphic with the group defined by the presentation. We show that there are model complexes of size in the order of the bit-complexity of the presentation that can be realized linearly in $4$-space. We further derive some applications of this result regarding embeddability problems in the euclidean $4$-space.