Geodesic
A note on a local combinatorial formula for the Euler class of a PL spherical fiber bundle
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXIII, Tome 507 (2021), pp. 35-58

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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. 
@article{ZNSL_2021_507_a3,
     author = {N. E. Mn\"ev},
     title = {A note on a local combinatorial formula for the {Euler} class of a {PL} spherical fiber bundle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {35--58},
     publisher = {mathdoc},
     volume = {507},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_507_a3/}
}
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N. E. Mnëv. A note on a local combinatorial formula for the Euler class of a PL spherical fiber bundle. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXIII, Tome 507 (2021), pp. 35-58. http://geodesic.mathdoc.fr/item/ZNSL_2021_507_a3/