Geodesic
Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III.~Moves $1\leftrightarrow5$ and Related Structures
Teoretičeskaâ i matematičeskaâ fizika, Tome 135 (2003) no. 2, pp. 179-195

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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 $3\to3$ 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.
Keywords: piecewise-linear manifolds - invariants of manifolds, Pachner moves, differential identities for Euclidean simplices
Mots-clés : acyclic complexes. 
@article{TMF_2003_135_2_a0,
     author = {I. G. Korepanov},
     title = {Euclidean {4-Simplices} and {Invariants} of {Four-Dimensional} {Manifolds:} {III.~Moves} $1\leftrightarrow5$ and {Related} {Structures}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {179--195},
     publisher = {mathdoc},
     volume = {135},
     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_135_2_a0/}
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I. G. Korepanov. Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III.~Moves $1\leftrightarrow5$ and Related Structures. Teoretičeskaâ i matematičeskaâ fizika, Tome 135 (2003) no. 2, pp. 179-195. http://geodesic.mathdoc.fr/item/TMF_2003_135_2_a0/