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We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. We first write formulas for moves 3 → 3 and 2 ↔ 4 based on the results in our two previous works and then study moves 1 ↔ 5 in detail. Based on this, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for the sphere S4; in particular, the complex does turn out to be acyclic.

Euclidean 4-simplices and invariants of four-dimensional manifolds: III. Moves 1 5 and related structures