Geodesic
Parameterizing the Simplest Grassmann–Gaussian Relations for Pachner Move 3–3
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider relations in Grassmann algebra corresponding to the four-dimensional Pachner move 3–3, assuming that there is just one Grassmann variable on each 3-face, and a 4-simplex weight is a Grassmann–Gaussian exponent depending on these variables on its five 3-faces. We show that there exists a large family of such relations; the problem is in finding their algebraic-topologically meaningful parameterization. We solve this problem in part, providing two nicely parameterized subfamilies of such relations. For the second of them, we further investigate the nature of some of its parameters: they turn out to correspond to an exotic analogue of middle homologies. In passing, we also provide the 2–4 Pachner move relation for this second case.
Keywords: four-dimensional Pachner moves; Grassmann algebras; Clifford algebras; maximal isotropic Euclidean subspaces. 
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     author = {Igor G. Korepanov and Nurlan M. Sadykov},
     title = {Parameterizing the {Simplest} {Grassmann{\textendash}Gaussian} {Relations} for {Pachner} {Move~3{\textendash}3}},
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}
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Igor G. Korepanov; Nurlan M. Sadykov. Parameterizing the Simplest Grassmann–Gaussian Relations for Pachner Move 3–3. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a52/

[1] Berezin F. A., The method of second quantization, Pure and Applied Physics, 24, Academic Press, New York, 1966 | MR | Zbl

[2] Berezin F. A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, 9, D. Reidel Publishing Co., Dordrecht, 1987 | MR | Zbl

[3] Björk J. E., Rings of differential operators, North-Holland Mathematical Library, 21, North-Holland Publishing Co., Amsterdam, 1979 | MR

[4] Cartier P., “Démonstration “automatique” d'identités et fonctions hypergéométriques (d'après {D}. {Z}eilberger)”, Séminaire Bourbaki 1991/92, Astérisque, 206, no. 746(3), 1992, 41–91 | MR | Zbl

[5] Chevalley C., Collected works, v. 2, The algebraic theory of spinors and {C}lifford algebras, Springer-Verlag, Berlin, 1997 | MR | Zbl

[6] Korepanov I. G., “Geometric torsions and an {A}tiyah-style topological field theory”, Theoret. and Math. Phys., 158 (2009), 344–354, arXiv: 0806.2514 | DOI | MR | Zbl

[7] Korepanov I. G., “Relations in {G}rassmann algebra corresponding to three- and four-dimensional {P}achner moves”, SIGMA, 7 (2011), 117, 23 pp., arXiv: 1105.0782 | DOI | MR | Zbl

[8] Korepanov I. G., Deformation of a $3\to 3$ {P}achner move relation capturing exotic second homologies, arXiv: 1201.4762

[9] Korepanov I. G., Special 2-cocycles and 3–3 {P}achner move relations in {G}rassmann algebra, arXiv: 1301.5581

[10] Korepanov I. G., Sadykov N. M., “Pentagon relations in direct sums and {G}rassmann algebras”, SIGMA, 9 (2013), 030, 16 pp., arXiv: 1212.4462 | DOI | MR | Zbl

[11] Lickorish W. B. R., “Simplicial moves on complexes and manifolds”, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999, 299–320, arXiv: math.GT/9911256 | DOI | MR | Zbl

[12] Pachner U., P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin., 12 (1991), 129–145 | MR | Zbl

[13] Petkovšek M., Wilf H. S., Zeilberger D., $A=B$, A K Peters Ltd., Wellesley, MA, 1996 | MR | Zbl