A circle bundle admits a semi-simplicial triangulation over a fixed base exactly when its Chern class has a binary 0/1 cocycle representative; on surfaces this bounds the Chern number by half the number of triangles.
Circle bundles
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Minimal triangulations of circle bundles, circular permutations and binary Chern cocycle
A circle bundle admits a semi-simplicial triangulation over a fixed base exactly when its Chern class has a binary 0/1 cocycle representative; on surfaces this bounds the Chern number by half the number of triangles.