Geodesic
Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. I, Tome 268 (2010), pp. 76-93.

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The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov–Gelfand–Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based on the notion of a universal local formula introduced by the author and on the usage of bistellar moves. We give a brief sketch for the first formula and a rather detailed exposition for the second one. For the second formula, we also succeed to simplify it by providing a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of two-dimensional combinatorial spheres into a linear combination of elementary cycles. 
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Alexander A. Gaifullin. Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. I, Tome 268 (2010), pp. 76-93. http://geodesic.mathdoc.fr/item/TM_2010_268_a6/

[1] Gabrielov A. M., Gelfand I. M., Losik M. V., “Kombinatornoe vychislenie kharakteristicheskikh klassov”, Funkts. analiz i ego pril., 9:2 (1975), 12–28 | MR | Zbl

[2] Gabrielov A. M., Gelfand I. M., Losik M. V., “Lokalnaya kombinatornaya formula dlya pervogo klassa Pontryagina”, Funkts. analiz i ego pril., 10:1 (1976), 14–17 | MR | Zbl

[3] MacPherson R., “The combinatorial formula of Gabrielov, Gelfand and Losik for the first Pontrjagin class”, Sèminaire Bourbaki 1976/77, Lect. Notes Math., 677, Springer, Berlin, 1978, Exp. 497, 105–124 | MR

[4] Gabrielov A. M., “Kombinatornye formuly dlya klassov Pontryagina i $GL$-invariantnye tsepi”, Funkts. analiz i ego pril., 12:2 (1978), 1–7 | MR | Zbl

[5] Gelfand I. M., MacPherson R. D., “A combinatorial formula for the Pontrjagin classes”, Bull. Amer. Math. Soc., 26:2 (1992), 304–309 | DOI | MR | Zbl

[6] Cheeger J., “Spectral geometry of singular Riemannian spaces”, J. Diff. Geom., 18:4 (1983), 575–657 | MR | Zbl

[7] Gaifullin A. A., “Lokalnye formuly dlya kombinatornykh klassov Pontryagina”, Izv. RAN. Ser. mat., 68:5 (2004), 13–66 | DOI | MR | Zbl

[8] Gaifullin A. A., “Postroenie kombinatornykh mnogoobrazii s zadannymi naborami linkov vershin”, Izv. RAN. Ser. mat., 72:5 (2008), 3–62 | DOI | MR | Zbl

[9] Levitt N., Rourke C., “The existence of combinatorial formulae for characteristic classes”, Trans. Amer. Math. Soc., 239 (1978), 391–397 | DOI | MR | Zbl

[10] Gaifullin A. A., “Vychislenie kharakteristicheskikh klassov mnogoobraziya po ego triangulyatsii”, UMN, 60:4 (2005), 37–66 | DOI | MR | Zbl

[11] Stiefel E., “Richtungsfelder und Fernparallelismus in $n$-dimensionalen Mannigfaltigkeiten”, Comment. Math. Helv., 8 (1936), 305–353 | DOI | MR | Zbl

[12] Whitney H., “On the theory of sphere bundles”, Proc. Nat. Acad. Sci. USA, 26 (1940), 148–153 | DOI | MR | Zbl

[13] Halperin S., Toledo D., “Stiefel–Whitney homology classes”, Ann. Math. Ser. 2, 96 (1972), 511–525 | DOI | MR | Zbl

[14] Cheeger J., “A combinatorial formula for Stiefel–Whitney classes”, Topology of manifolds, Proc. Univ. Georgia, 1969, Markham Publ., Chicago, 1970, 470–471

[15] Atiyah M. F., Patodi V. K., Singer I. M., “Spectral asymmetry and Riemannian geometry”, Bull. London Math. Soc., 5:2 (1973), 229–234 | DOI | MR | Zbl

[16] Rurk K., Sanderson B., Vvedenie v kusochno lineinuyu topologiyu, Mir, M., 1974 | MR

[17] Whitehead J. H. C., “Note on manifolds”, Quart. J. Math. Oxford Ser., 12 (1941), 26–29 | DOI | MR | Zbl

[18] Cairns S. S., “Isotopic deformations of geodesic complexes on the 2-sphere and on the plane”, Ann. Math. Ser. 2, 45:2 (1944), 207–217 | DOI | MR | Zbl

[19] Ho C.-W., “On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I”, Trans. Amer. Math. Soc., 181 (1973), 213–233 | DOI | MR | Zbl

[20] Pachner U., “Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten”, Abh. Math. Sem. Univ. Hamburg., 57 (1987), 69–86 | DOI | MR | Zbl

[21] Pachner U., “P. L. homeomorphic manifolds are equivalent by elementary shellings”, Eur. J. Combin., 12:2 (1991), 129–145 | DOI | MR | Zbl

[22] Kazarian M. È., “The Chern–Euler number of circle bundle via singularity theory”, Math. scand., 82 (1998), 207–236 | MR | Zbl

[23] Kühnel W., Banchoff T. F., “The 9-vertex complex projective plane”, Math. Intell., 5:3 (1983), 11–22 | DOI | MR | Zbl

[24] Björner A., Lutz F. H., “Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere”, Exp. Math., 9:2 (2000), 275–289 | DOI | MR | Zbl