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

TL;DR

This paper explores explicit combinatorial formulas for the first Pontryagin class, comparing classical and new local approaches, and introduces a simplified algorithm for cycle decomposition in bistellar move graphs.

Contribution

It presents a new simplified algorithm for decomposing cycles in bistellar move graphs, enhancing the combinatorial computation of Pontryagin classes.

Findings

The classical Gabrielov-Gelfand-Losik formula based on configuration spaces.

A new local combinatorial formula using bistellar moves.

A simplified cycle decomposition algorithm for bistellar move graphs.

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.