Gaifullin Alexander A., Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class [Recensione]

The problem of finding explicit combinatorial formulae for the Pontryagin classes of triangulated manifolds was originally faced by Gabrielov, Gelfand and Losik in [Funct. Anal. Appl. 9, 103-115 (1975; Zbl 0312.57016)], and successively taken into account by various authors: see Russ. Math. Surv. 60, No. 4, 615-644 (2005; Zbl 1139.57026)] for a survey on the topic, with a comparison among different formulae. The present paper discusses the only two known combinatorial formulae for the first Pontryagin class that can be used for real computation, i.e. the classical Gabrielov-Gelfand-Losik formula and the local formula obtained by the author in [Izv. Math. 68, No. 5, 861-910 (2004; Zbl 1068.57022)]. Note that the first formula is presented not according to the original approach, based on endowing a triangulated manifold with locally flat connections, but according to MacPherson’s approach, based on the construction of a homology Gaussian mapping for a combinatorial manifold: see [Semin. Bourbaki, Vol. 1976/77, Lect. Notes Math. 677, 105-124 (1978; Zbl 0388.57013)]. A detailed exposition of the second formula is provided, together with the related notions and results concerning bistellar moves (see [Abh. Math. Semin. Univ. Hamb. 57, 69-86 (1987; Zbl 0651.52007)] and [Eur. J. Comb. 12, No.2, 129-145 (1991; Zbl 0729.52003)]) and the existence and uniqueness of universal local formulae for polynomials in rational Pontryagin classes. In the present paper the author succeeds in considerably simplifying his own explicit formula for the first Pontryagin class, by giving 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.