Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 021, 7 pages      math.GT/0510095

Pachner Move 3 –> 3 and Affine Volume-Preserving Geometry in R3

Igor G. Korepanov
South Ural State University, 76 Lenin Ave., 454080 Chelyabinsk, Russia

Received October 06, 2005, in final form November 21, 2005; Published online November 24, 2005

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.

Key words: piecewise-linear topology; Pachner move; algebraic relation; three-dimensional affine geometry.

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  1. Lickorish W.B.R., Simplicial moves on complexes and manifolds, Geom. Topol. Monographs, 1999, Vol. 2, Proceedings of the Kirbyfest, 299-320, math.GT/9911256.
  2. Diatta A., Medina A., Classical Yang-Baxter equation and left invariant affine geometry on Lie groups. Manuscripta mathematica, 2004, V.114, N 4, 477-486, math.DG/0203198.
  3. Korepanov I.G., A formula with volumes of five tetrahedra and discrete curvature, nlin.SI/0003001.
  4. Korepanov I.G., A formula with hypervolumes of six 4-simplices and two discrete curvatures, nlin.SI/0003024.
  5. Korepanov I.G., Invariants of PL manifolds from metrized simplicial complexes. Three-dimensional case, J. Nonlinear Math. Phys., 2001, V.8, N 2, 196-210, math.GT/0009225.
  6. Korepanov I.G., Martyushev E.V., A classical solution of the pentagon equation related to the group SL(2), Theor. Math. Phys., 2001, V.129, N 1, 1320-1324.
  7. Ponzano G., Regge T., Semi-classical limit of Racah coefficients, in "Spectroscopic and Group Theoretical Methods in Physics", Editor F. Bloch, North-Holland, 1968, 1-58.
  8. Roberts J., Classical 6j-symbols and the tetrahedron, Geom. Topol., 1999, V.3, 21-66, math-ph/9812013.
  9. Taylor Y., Woodward C., Spherical tetrahedra and invariants of 3-manifolds, math.GT/0406228.
  10. Korepanov I.G., SL(2)-Solution of the pentagon equation and invariants of three-dimensional manifolds, Theor. Math. Phys., 2004, V.138, N 1, 18-27, math.AT/0304149.
  11. Korepanov I.G., Euclidean 4-simplices and invariants of four-dimensional manifolds: I. Moves 3 3, Theor. Math. Phys., 2002, V.131, N 3, 765-774, math.GT/0211165.
  12. Korepanov I.G., Euclidean 4-simplices and invariants of four-dimensional manifolds: II. An algebraic complex and moves 2" 4, Theor. Math. Phys., 2002, V.133, N 1, 1338-1347, math.GT/0211166.
  13. Korepanov I.G., Euclidean 4-simplices and invariants of four-dimensional manifolds: III. Moves 1" 5 and related structures, Theor. Math. Phys., 2003, V.135, N 2, 601-613, math.GT/0211167.
  14. Atiyah M.F., Topological quantum field theory, Publications Mathématiques de l'IHÉS, 1988, V.68, 175-186.

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