  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.
