Note on local combinatorial formula for Euler class of PL spherical fiber bundle

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