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Gelʹfand, I. M. and MacPherson, R. D. "A combinatorial formula for the Pontrjagin classes" Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 304–309.

In the above paper the authors claimed a construction of combinatorial formula for the Pontrjagin classes. According to mathscinet review: MR1129313, it seems the construction is not really local. So the existence of such formula is still open?

Also for first Pontrjagin class, do I understand correctly that "Gabrièlov, A. M.; Gelʹfand, I. M.; Losik, M. V. A local combinatorial formula for the first Pontrjagin class" is a well know result? If so is there any English paper gave detailed constructions?

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I believe there is a formula like the one you seek together with a good survey of the previous literature in the paper "Local formulae for combinatorial Pontrjagin classes" by Gaifullin, available here.

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  1. That the Gabrielov, Gelfand Losik paper is "well-known" you can see from Mathscinet citation records.

  2. This journal is translated into English.

  3. Here are some other papers: MR1129313, MR0410758.

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