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We consider computational complexity of problems related to the fundamentalgroup and the first homology group of (embeddable) $2$-complexes. We show, asan extension of an earlier work, that computing first homology of $2$-complexesis equivalent in computational complexity to matrix diagonalization. That is,the usual procedures for computing homology cannot be improved other than bymatrix 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 grouppresentations, called model $2$-complexes. Model complexes have fundamentalgroup isomorphic with the group defined by the presentation. We show that thereare model complexes of size in the order of the bit-complexity of thepresentation that can be realized linearly in $4$-space. We further derive someapplications of this result regarding embeddability problems in the euclidean$4$-space.

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