Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: II. An Algebraic Complex and Moves $2\leftrightarrow 4$

We present sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex determining a triangulation of a manifold. If a sequence is an acyclic complex, then we can construct a manifold invariant using its torsion. We demonstrate this first for three-dimensional manifolds and then construct the part of this program for four-dimensional manifolds pertaining to moves $2\leftrightarrow 4$.