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The combinatorial formula of Gabrielov, Gelfand and Losik for the first Pontrjagin class

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The combinatorial formula of Gabrielov, Gelfand and Losik for the first Pontrjagin class SÉMINAIRE N. BOURBAKI ROBERTMACPHERSON The combinatorial formula of Gabrielov, Gelfand and Losik for the first Pontrjagin class Séminaire N. Bourbaki, 1976-1977, exp. no 497, p. 105-124. <http://www.numdam.org/item?id=SB_1976-1977__19__105_0> © Association des collaborateurs de Nicolas Bourbaki, 1976-1977, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisa- tion (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 105 Seminaire BOURBAKI 29e annee, 1976/77, n° 497 Fevrier 1977 THE COMBINATORIAL FORMULA OF GABRIELOV, GELFAND AND LOSIK FOR THE FIRST PONTRJAGIN CLASS by Robert MacPHERSON The problem addressed here is to find a formula for the Pontrjagin classes of a polyhedral manifold X in terms of the combinatorial structure of X . The exis- tance of these classes was first established by Thom [18] by a nonconstructive argu- ment. One motivation for wanting an explicit formula is the hope of extending the deep results on signatures of elliptic operators which relate to the curvature for- mula for the Pontrjagin classes to the framework of difference operators on polyhe- dra [16]. Another motivation is the question of whether there exists a purely local formula for the Pontrjagin numbers i.e. a formula using only the set of stars of vertices of X, not how they are put together. It is known that no topological inva- riants other than the Pontrjagin numbers or the Euler characteristic can be given by a purely local formula [10]. Combinatorial formulas exist for the other characteristic cla

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