Geodesic
Minimal triangulations of circle bundles, circular permutations, and the binary Chern cocycle
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 87-107

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We investigate a PL topology question: which circle bundles can be triangulated over a given triangulation of the base? The question gets a simple answer emphasizing the role of minimal triangulations encoded by local systems of circular permutations of vertices of the base simplices. The answer is based on an experimental fact: the classical Huntington transitivity axiom for cyclic orders can be expressed as the universal binary Chern cocycle. 
@article{ZNSL_2019_481_a6,
     author = {N. Mn\"ev},
     title = {Minimal triangulations of circle bundles, circular permutations, and the binary {Chern} cocycle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {87--107},
     publisher = {mathdoc},
     volume = {481},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a6/}
}
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N. Mnëv. Minimal triangulations of circle bundles, circular permutations, and the binary Chern cocycle. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 87-107. http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a6/