Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III. Moves $1\leftrightarrow5$ and Related Structures

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 $3\to3$ and $2\leftrightarrow4$ based on the results in our two previous works and then study moves $1\leftrightarrow5$ 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 $S^4$; in particular, the complex does turn out to be acyclic.