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

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