A note on a local combinatorial formula for the Euler class of a PL spherical fiber bundle

TL;DR

This paper introduces a local combinatorial formula for calculating the Euler class of PL spherical fiber bundles using a chain of subdivisions and a matrix Hodge theory twisting cochain, linking combinatorics with topological invariants.

Contribution

It provides a novel combinatorial (matrix-based) formula for the Euler class of PL spherical fiber bundles, connecting subdivision chains with topological invariants.

Findings

The formula computes the Euler class as a rational number.

It relates combinatorial subdivisions to topological invariants.

The approach uses a Hodge theory twisting cochain in homology.

Abstract

We present a local combinatorial formula for the Euler class of a $n$-dimen\-si\-onal PL spherical fiber bundle as a rational number $e_{\it CH}$ associated to a chain of $n+1$ abstract subdivisions of abstract $n$-spherical PL cell complexes. The number $e_{\it CH}$ is a combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirsch's homology model of the bundle associated with PL combinatorics of the bundle.