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

Abstract

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.

18 pages, 4 LaTeX pseudofigures, Talk at conference dedicated to the Centennial Anniversary of L.S. Pontryagin (Moscow, 2008), to appear in Proc. Steklov Math. Institute

References in corpus (1)

• Construction of combinatorial manifolds with the prescribed sets of links of vertices

Cited by in corpus (3)

• A 15-vertex triangulation of the quaternionic projective plane
• 634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
• New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane