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

SIGMA 7 (2011), 117, 23 pages      arXiv:1105.0782

Relations in Grassmann Algebra Corresponding to Three- and Four-Dimensional Pachner Moves

Igor G. Korepanov
Moscow State University of Instrument Engineering and Computer Sciences, 20 Stromynka Str., Moscow 107996, Russia

Received May 15, 2011, in final form December 16, 2011; Published online December 18, 2011

New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally – using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated boundary, and present example calculations confirming its nontriviality.

Key words: Pachner moves; Grassmann algebras; algebraic topology.

pdf (507 kb)   tex (53 kb)


  1. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model, Classical Quantum Gravity 26 (2009), 155014, 48 pages, arXiv:0803.3319.
  2. Berezin F.A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, Vol. 9, D. Reidel Publishing Company, Dordrecht, 1987.
  3. Bel'kov S.I., Korepanov I.G., A matrix solution of the pentagon equation with anticommuting variables, Theoret. and Math. Phys. 163 (2010), 819-830, arXiv:0910.2082.
  4. Bel'kov S.I., Korepanov I.G., Martyushev E.V., A simple topological quantum field theory for manifolds with triangulated boundary, arXiv:0907.3787.
  5. Dubois J., Korepanov I.G., Martyushev E.V., A Euclidean geometric invariant of framed (un)knots in manifolds, SIGMA 6 (2010), 032, 29 pages, math.GT/0605164.
  6. Fomenko A.T., Matveev S.V., Algorithmic and computer methods for three-manifolds, Mathematics and its Applications, Vol. 425, Kluwer Academic Publishers, Dordrecht, 1997.
  7. GAP - Groups, Algorithms, Programming - a system for computational discrete algebra,
  8. Korepanov A.I., Korepanov I.G., Sadykov N.M., PL: Piecewise-linear topology using GAP,
  9. Korepanov I.G., Algebraic relations with anticommuting variables for four-dimensional Pachner moves 3→3 and 2↔4, arXiv:0911.1395.
  10. Korepanov I.G., Two deformations of a fermionic solution to pentagon equation, arXiv:1104.3487.
  11. Korepanov I.G., Sadykov N.M., Four-dimensional Grassmann-algebraic TQFT's, work in progress.
  12. Lickorish W.B.R., Simplicial moves on complexes and manifolds, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., Vol. 2, Geom. Topol. Publ., Coventry, 1999, 299-320, math.GT/9911256.
  13. Martyushev E.V., Euclidean simplices and invariants of three-manifolds: a modification of the invariant for lens spaces, Izv. Chelyabinsk. Nauchn. Tsentra 2003 (2003), no. 2 (19), 1-5, math.AT/0212018.
  14. Martyushev E.V., Euclidean geometric invariants of links in 3-sphere, Izv. Chelyabinsk. Nauchn. Tsentra 2004 (2004), no. 4 (26), 1-5, math.GT/0409241.
  15. Maxima, a computer algebra system,
  16. Pachner U., P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), 129-145.
  17. Turaev V.G., Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.

Previous article   Next article   Contents of Volume 7 (2011)