This article is cited in 9 papers

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

I. G. Korepanov

South Ural State University

Abstract: 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, acyclic complexes.

Received: 17.05.2002

DOI: 10.4213/tmf185

 English version: Theoretical and Mathematical Physics, 2003, 135:2, 601–613

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