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We construct a family of heptagon relations -- algebraic imitations of five-dimensional Pachner move 3--4, parameterized by simplicial 3-cocycles., Comment: 11 pages

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geometric values such as volumes of different dimensions and various kinds of angles. Then we construct an acyclic complex made of differentials of these geometric values, and its torsion will lead, depending on the specific kind of this complex, to some manifold or knot invariants. In this paper, we limit ourselves to constructing a simplest kind of acyclic complex, from which a three-manifold invariant can be obtained., Comment: 18 pages, 2 figures

We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.

What discuss the problem of obtaining new manifold invariants via different analogues of 6j-symbols and the torsion of acyclic complexes., Comment: 3 pages

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. First, we write out formulas for moves 3 -> 3 and 2 <-> 4 based on the results of our two previous works, and then we study in detail moves 1 <-> 5. On this basis, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for sphere S^4; in particular, the complex turns out, indeed, to be acyclic., Comment: LaTeX2e, 18 pages

We write out some 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 - a triangulation of a manifold. If the sequence turns out to be an acyclic complex then one can construct a manifold invariant out of its torsion. We demonstrate this first for three-dimensional manifolds, and then we conduct the part, related to moves 2 <-> 4, of the corresponding work for four-dimensional manifolds., Comment: LaTeX2e, 13 pages

We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on the usage of Euclidean geometric values., Comment: LaTeX2e, 12 pages

An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.