SL(2)-solution of the pentagon equation and invariants of three-dimensional manifolds

Building on a classical solution to the pentagon equation, constructed earlier by the author and E.V. Martyushev and related to the flat geometry invariant under the group SL(2), we construct an algebraic complex corresponding to a triangulation of a three-manifold. In case if this complex is acyclic (which is confirmed by examples), we use it for constructing a manifold invariant .Comment: 13 pages; accepted for publication in Theor. Math. Phy

Pachner Move 3->3 and Affine Volume-Preserving Geometry in R^3

Pachner move 3 ->3 deals with triangulations of four-dimensional manifolds. We present an algebraic relation corresponding in a natural way to this move and based, a bit paradoxically, on three-dimensional geometry.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA