Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class

Abstract

Abstract. The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-GelfandLosik 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.

A 15-Vertex Triangulation of the Quaternionic Projective Plane

“…This algorithm was found by Gaifullin [14], but some subcases were missed out. In the present article we eliminate the gap, thereby the algorithm of decomposition is now complete.…”

“…In 2004 Gaifullin [12] (cf. [13,14]) constructed an explicit algorithm for computing the first rational Pontryagin class of a combinatorial manifold. A combinatorial manifold of dimension n is a simplicial complex K, such that the link of any vertex of K is PL homeomotphic to the boundary of the n-dimensional simplex.…”

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

“…Gorodkov's result in the quaternionic case (see [20], [21]) was based on the computation of the first Pontryagin class of the Brehm-Kühnel combinatorial manifolds using an explicit combinatorial formula due to the author of the present paper, cf. [14], [15], [17], [19]. However, to decide whether a 16-manifold like a projective plane is indeed homeomorphic to OP 2 one needs to compute the second Pontryagin class and the first exotic PL characteristic class of it.…”

How many simplices are needed to triangulate a Grassmannian?

“…However, with an explicit triangulation at hand one can go even further and use computers to answer other specific questions about the topology of the manifold. For example, there is a long-standing problem concerning the computation of rational Pontrjagin classes of combinatorial manifolds, see Gelfand-MacPherson [7] and Gaifullin [6]. In particular Gaifullin [6] described an explicit algorithm for the computation of the first Pontrjagin class of a manifold.…”

“…For example, there is a long-standing problem concerning the computation of rational Pontrjagin classes of combinatorial manifolds, see Gelfand-MacPherson [7] and Gaifullin [6]. In particular Gaifullin [6] described an explicit algorithm for the computation of the first Pontrjagin class of a manifold. The computation uses only the combinatorial structure of the triangulation and does not need any additional data.…”